R Package ‘stratifyR’

Type Package

Title Optimal Stratification of Univariate Populations

Version 2.0-1

Date 2026-06-04

Author Karuna G. Reddy, M.G.M. Khan

Maintainer Karuna G. Reddy <karuna.reddy@auckland.ac.nz>

Description This implements the stratification of univariate populations under stratified sampling designs using the method of Khan et al. (2002) https://doi.org/10.1177/0008068320020518, Khan et al. (2008) https://www150.statcan.gc.ca/n1/pub/12-001-x/2008002/article/10761-eng.pdf and Khan et al. (2015) https://doi.org/10.1080/02664763.2015.1018674. It determines the Optimum Strata Boundaries (OSB) and Optimum Sample Sizes (OSS) for the study variable, \(y\), using the best-fit frequency distribution of a survey variable (if data is available) or a hypothetical distribution (if data is not available). The method formulates the problem of determining the OSB as mathematical programming problem which is solved by using a dynamic programming technique. If a dataset of the population is available to the surveyor, the method estimates its best-fit distribution and determines the OSB and OSS under Neyman allocation directly. When the dataset is not available, stratification is made based on the assumption that the values of the study variable, \(y\), are available as hypothetical realizations of proxy values of \(y\) from recent surveys. Thus, it requires certain distributional assumptions about the study variable. At present, it handles stratification for the populations where the study variable follows a continuous distribution, namely, Pareto, Triangular, Right-triangular, Weibull, Gamma, Exponential, Uniform, Normal, Log-normal and Cauchy distributions. In this vignette, the two major functionalities in the package are applied to a number of real and simulated populations and to some hypothetical populations.

License GPL (>= 2)

LazyData yes

NeedsCompilation yes

Depends R (>= 4.4.2), MASS, fitdistrplus, actuar, triangle, mc2d, zipfR

Repository CRAN

Date/Publication 2026-06-04 10:00:00

Using stratifyR through the interactive app

The quickest and easiest way to use stratifyR is through its built-in interactive application. The app gives you the complete workflow — load your own data or pick a built-in dataset, compute the Optimum Strata Boundaries (OSB) and Optimum Sample Sizes (OSS) with the DP, COBYLA or GLOBAL solvers, compare design efficiencies, explore the boundaries visually, and download the results — entirely through a point-and-click interface, with no R programming required. For most users, and especially for anyone who wants results quickly, this is the recommended way to work with the package.

To launch it, install the app’s helper packages once (if you do not already have them) and then call stratifyRApp():

# One-time install of the app's helper packages (skip any you already have)
install.packages(c("shiny", "bslib", "DT"))

# Optional extras: interactive 3-D / slider plots and Excel (.xlsx) import
install.packages(c("plotly", "readxl"))

# Launch the app
library(stratifyR)
stratifyRApp()

The application opens in your web browser (or as a standalone desktop-style window). The remainder of this vignette documents the underlying R functions, for readers who prefer to script their analyses or embed stratifyR in a larger workflow.

1 Introduction

The main aim of stratification is to produce estimators with a small variance when a population characteristic \((y)\) is under study. A simple method that can be used to create strata for this population, if \(y\) itself is the stratification variable. The ideal situation is that the distribution of such a study variable is known and the OSB can be determined by placing boundaries on the range of this distribution at suitable cut-points. This problem of determining the OSB, when both the estimation and stratification variables are the same, was first discussed by Dalenius (1950). He provided equations for the determination of stratum boundaries that minimize the variance of population estimates under optimal allocation. Dalenius (1957) futher proposed a solution to the problem by taking equal intervals of the cumulative square root of frequency scale of the stratification variable.

One of the many kinds of stratification methods that has been proposed in the literature is due to Bühler and Deutler (1975). They formulated the problem of determining OSB as an optimization problem and developed a computational technique to solve the problem by using Dynamic Programming (DP). A good review of this method can be found in Khan et al. (2008). The procedure is also applied by Khan et al. (2002), Khan et al. (2003), Khan et al. (2005), Khan et al. (2008), Khan et al. (2009), Khan, Rao, et al. (2015), Khan, Reddy, et al. (2015) for determining OSB for many different distributions. With the known frequency function of the study variable, they considered the problem of finding OSB as an equivalent problem of determining Optimum Strata Width (OSW), which is formulated as a Mathematical Programming Problem (MPP) and solved by using DP technique. They applied the technique to several univariate populations where the study variables followed different probability distributions. In this package, a univariate stratification technique is developed, which is based on the probability distribution assumed by the stratification variable as discussed by the authors above.

As implemented in this package, there are many advantages of using the method. The real advantage of the stratifyR package is that when the dataset of the study variable is not available, which may occur in practice, the package is still able to construct OSB and OSS based on the distributional assumption on the data. Moreover, the population survey can be a costly and time-consuming affair, hence, this approach also has the advantage of determining OSB for the study variable that is not available prior to conducting the survey. This, of course, requires certain distributional or parametric assumptions on the study variables, which can easily be obtained from recent or past surveys.

Other advantages of the method are that it leads to substantial gains in the precision of the estimates over other available methods. Results reveal that the variances get smaller with increasing number of strata \((L)\), and they get much smaller at a much faster rate than other available methods. Once the OSB have been determined, the Optimum Sample Sizes (OSS) can be easily calculated for each stratum using Neyman allocation (Neyman (1934)). The algorithm may be a little slow, however, it does provide very efficient results which is probably the most important objective of our survey estimation efforts.

2 The Package in a Glance

The stratifyR package implements the DP technique (from various literature by Khan et al) as a stratification procedure for univariate populations, when the stratification variable follows a continuous probability distribution, namely: uniform, triangular, right-triangular, pareto, exponential, normal, log-normal, cauchy, weibull and gamma. The package is able to determine the OSB and OSS from real data and as well as hypothetical situations when the dataset is not available. The latter requires some assumptions about the distribution and its initial value, range, parameter values, fixed sample size, etc.

There are two major functions which basically solve the two types of stratification problems: strata.data() which carries out univariate stratification for those univariate populations where dataset is available and strata.distr() which performs stratification when dataset is not available prior to conducting the survey.

In the former case, data on the study variable, number of strata \((h)\), fixed sample size \((n)\) and population size \((N)\) are used as the input arguments to the strata.data() function in the package. In the latter case, strata.distr() function is called which requires the distribution to be assumed, its parameters, the inital value and the estimated range of the distribution, fixed sample \((n)\) and population sizes \((N)\). When executed, both the functions output the OSB and OSS, amongst other quantities such as stratum weight \((W_{h})\), stratum variance \((S_{h}^{2})\), stratum objective function values \((W_{h}S_{h})\), stratum sample sizes \((n_{h})\), stratum population sizes \((N_{h})\) and stratum sampling fraction \((f_{h})\).

The following sections show the general formulation of the problem of stratification, the DP solution procedure, the concept of Neyman allocation as the method of determining sample size and the overview of package functionalities. The package is then applied to numerous survey populations with real and simulated data to illustrate its application.

3 General Formulation of the Univariate Stratification Problem

Let the target population of the variable under study be stratified into \(L\) strata where the estimation of the mean of the study variable \((y)\) is of interest. If a simple random sample of size \(n_{h}\) is to be drawn from \(h^{th}\) stratum with sample mean \(\bar{y}_{h}\), then the stratified sample mean, \(\bar{y}_{st}\), is given by \[\begin{eqnarray} \bar{y}_{st}=\sum_{h=1}^{L}W_{h}\bar{y}_{h}, \tag{1} \end{eqnarray}\] where \(W_{h}\) (stratum weight) is the proportion of the population contained in the \(h^{th}\) stratum.

When the finite population correction factors are ignored, under the Neyman (1934) allocation, the variance of \(\bar{y}_{st}\) is given by \[\begin{eqnarray} Var(\bar{y}_{st})=\dfrac{\left(\sum_{h=1}^{L}W_{h}S_{h}\right)^{2}}{n}, \tag{2} \end{eqnarray}\] where \(S_{h}^{2}\) is the stratum variance for the study variable in the \(h^{th}\) (where \(h=1, 2, ..., L\)) stratum and \(n\) is the preassigned total sample size.

Let \(f(y);\;a\leq y\leq b\) be the frequency function of the study variable, \(y\), on which OSB are to be constructed. If the population mean of this study variable is estimated under Neyman allocation, then the problem of determining OSB is to cut up the range, \(d=b-a\), at \((L-1)\) intermediate points \(a=y_{0} \leq y_{1} \leq y_{2} \leq, ..., \leq y_{L-1} \leq y_{L}=b\,\) such that (2) is minimum. The lower and upper bounds of the study variable are denoted by \(a\) and \(b\) respectively and the cut-points \(y_{0}, y_{1}, y_{2}, ..., y_{L-1}\) are the OSB.

For a fixed sample size \(n\), minimizing the expression of the right hand side of equation (2) is equivalent to minimizing \[\begin{eqnarray} \sum_{h=1}^{L} W_{h}S_{h} \tag{3} \end{eqnarray}\]

If \(f(y)\) is known and integrable, the stratum weight \((W_{h})\), stratum variance \((S_{h}^{2})\) and stratum mean \((\mu_{h})\) can be obtained as a function of the boundary points \(y_{h}\) and \(y_{h-1}\) by using the following expressions: \[\begin{eqnarray}\label{stratumweight} W_{h} = \int_{y_{h-1}}^{y_{h}} f(y)dy \tag{4} \end{eqnarray}\]

\[\begin{eqnarray} S_{h}^{2}= \dfrac{1}{W_{h}}\int_{y_{h-1}}^{y_{h}}y^{2}f(y)dy - \mu_{h}^{2} \tag{5} \end{eqnarray}\]

\[\begin{eqnarray} \textrm{where}\;\;\;\mu_{h}=\dfrac{1}{W_{h}}\int_{y_{h-1}}^{y_{h}}yf(y)dy \tag{6} \end{eqnarray}\]

where \((y_{h-1}, y_{h})\) are the boundaries of \(h^{th}\) stratum.

Thus, the objective function in (3) could be expressed as a function of boundary points \(y_{h}\) and \(y_{h-1}\) only. We further define \[\begin{eqnarray} l_{h}=y_{h}-y_{h-1}; \; h=1,2, ..., L \tag{7} \end{eqnarray}\]

where \(l_{h}\geq 0\) denotes the range or width of the \(h^{th}\) stratum. Then, the range of the distribution, \(d = b - a\), is expressed as a function of stratum width as: \[\begin{eqnarray}\label{215} \sum_{h=1}^{L}l_{h}=\sum_{h=1}^{L}(y_{h}-y_{h-1}) = b-a = y_{L} - y_{0} = d \tag{8} \end{eqnarray}\] The \(h^{th}\) stratification point \(y_{h};\;h=1,2,..., L\,\) is then expressed as \(y_{h}=y_{h-1}+l_{h}\) and from (8), the problem can be treated as an equivalent problem of determining Optimum Strata Widths (OSW), \(l_{1}, l_{2}, ..., l_{L}\). Due to the special nature of functions, the problem may be treated as a function of \(l_{h}\) alone and can be expressed as: \[\begin{eqnarray}\label{genMPP} &\textrm{Minimize}& \;\;\; \sum_{h=1}^{L}\phi_{h}(l_{h}),\nonumber \\ &\textrm{subject to}&\;\;\;\; \sum_{h=1}^{L}l_{h} = d,\nonumber \\ &\textrm{and}& \;\;\;\; l_{h} \geq 0; \;\; h = 1,2,...,L. \tag{9} \end{eqnarray}\]

4 Dynamic Programming Technique as a Solution Procedure

Many multistage decision problems can be formulated as a Mathematical Programming Problem (MPP). The Dynamic Programming technique, developed by Bellman, R E (1957), is a computational method which is well suited for solving MPPs that may be treated as a multistage decision problem. The DP technique determines the optimum solution of a multistage problem by decomposing it into stages, each stage comprising of a single stage. The advantage of the decomposition is that the optimization process at each stage involves one variable only, which simplifies the computational task by dealing with all variables simultaneously.

The solution to an MPP is achieved in a sequential manner starting from one stage problem, moving onto a two stage problem, to a three stage problem and so on until finally all stages are included. The solution for \(n\) stages is obtained by adding the \(n^{th}\) stage to the solution of \(n - 1\) stages.

The basic concept of DP technique is contained in the principle of optimality proclaimed by Bellman, R E (1957), which implies that given the initial state of a system, an optimal policy for the subsequent stages does not depend upon the policy adopted at the preceding stages. It determines the optimum solution of a multi-variable problem by decomposing it into stages, each stage compromising a single variable sub-problem. A dynamic programming model is basically a recursive equation which links the different stages of the problem in a manner which guarantees that each stage’s optimal feasible solution is also optimum and feasible for the entire problem.

Consider the following sub-problem of (9) for first \(k(<L)\) strata:

\[\begin{eqnarray} &\text{Minimize}& \;\;\;\; \sum_{h=1}^{k}\phi_{h}(l_{h}),\nonumber \\ &\text{subject to}&\;\;\;\;\; \sum_{h=1}^{k}l_{h} = d_{k},\nonumber \\ &\text{and}& \;\;\;\;\; l_{h} \geq 0; \;\; h = 1,2,...,k.\tag{10} \end{eqnarray}\]

where \(d_{k}< d\) is the total width available for division into strata or the state value at stage \(k\). Note that \(d_{k} = d\) for \(k = L\).

The transformation functions are given by:

\[\begin{eqnarray*} d_{k}\;\;&=&\;\;l_{1}+l_{2}+...+l_{k},\\ d_{k-1}\;\;&=&\;\;l_{1}+l_{2}+...+l_{k-1}\;=\;d_{k}-l_{k},\\ d_{k-2}\;\;&=&\;\;l_{1}+l_{2}+...+l_{k-2}\;=\;d_{k-1}-l_{k-1},\\ &\vdots&\;\;\;\;\;\;\;\;\;\;\;\;\vdots\\ d_{2}\;\;&=&\;\;l_{1}+l_{2}\;=\;d_{3}-l_{3},\\ d_{1}\;\;&=&\;\;l_{1}\;=\;d_{2}-l_{2}. \end{eqnarray*}\]

Let \(\Phi_{k}(d_{k})\) denote the minimum value of the objective function of MPP (10), that is,

\[\begin{equation} \Phi_{k}(d_{k}) = \text{min}\left[ \sum_{h=1}^{k}\phi_{h}(l_{h})\bigg|\sum_{h=1}^{k}l_{h}=d_{k}, \,\textrm{and}\,l_{h}\;\geq 0;\,h=1,2,...,k \;\textrm{and}\; 1\leq k \leq L \right].\nonumber \end{equation}\]

With the above definition of \(\Phi_{k}(d_{k})\), (10) is equivalent to finding \(\Phi_{L}(d)\) recursively by finding \(\Phi_{k}(d_{k})\) for \(k = 1, 2, ..., L\) and \(0 \leq d_{k} \leq d.\)

We can write:

\[\begin{equation} \Phi_{k}(d_{k}) = \text{min}\left[\phi_{k}(l_{k})+ \sum_{h=1}^{k-1}\phi_{h}(l_{h})\bigg|\sum_{h=1}^{k-1}l_{h}=d_{k}-l_{k}, \;\textrm{and}\;l_{h}\;\geq\;0;\;h=1,2,...,k\right].\nonumber \end{equation}\]

For a fixed value of \(l_{k}\); \(0 \leq l_{k} \leq d_{k}\),

\[\begin{equation} \Phi_{k}(d_{k}) = \phi_{k}(l_{k})+ \text{min}\left[ \sum_{h=1}^{k-1}\phi_{h}(l_{h})\bigg|\sum_{h=1}^{k-1}l_{h}=d_{k}-l_{k}, \;\textrm{and}\;l_{h}\;\geq 0;\;h=1,2,...k-1\;\textrm{and}\;\; 1\leq k \leq L\right].\nonumber \end{equation}\]

Using the Bellman’s principle of optimality, a forward recursive equation can be written as:

\[\begin{equation} \Phi_{k}(d_{k}) = {\text{min} \atop 0 \leq l_{k} \leq d_{k}}\left[\phi_{k}(l_{k}) + \Phi_{k-1}(d_{k}-l_{k})\right],\;\;k\;\geq\;2.\tag{11} \end{equation}\]

For the first stage, that is, for \(k=1\):

\[\begin{equation} \Phi_{1}(d_{1}) = \phi_{1}(d_{1})\; \Longrightarrow\;\;l_{1}^{*}=d_{1},\tag{12} \end{equation}\]

where \(l_{1}^{*}=d_{1}\) is the optimum width of the first stratum. The relations (11) and (12) are solved recursively for each \(k=1, 2, ..., L\,\) and \(0 \leq d_{k} \leq d\), and \(\Phi_{L}(d)\) is obtained. From \(\Phi_{L}(d)\) the optimum width of \(L^{th}\) stratum, \(l_{L}^{*}\), is obtained. From \(\Phi_{L-1}(d - l_{L}^{*})\) the optimum width of \((L - 1)^{th}\) stratum, \(l_{L-1}^{*}\), is obtained and so on until \(l_{1}^{*}\) is obtained.

The above algorithm of the Dynamic Programming solution procedure to solve the MPP given in (9) is summarized with the following steps:

1. Start at \(k=1\). Set \(\Phi_{0}(d_{0})=0\).
2. Calculate \(\Phi_{1}(d_{1})\), the minimum value of RHS of \((12)\) for \(l_{1}=d_{1},\; 0\leq l_{1}\leq d_{1}\), and \(0\leq d_{1}\leq d\).
3. Record \(\Phi_{1}(d_{1})\) and \(l_{1}\).
4. For \(k=2\), express the state variable as \(d_{k-1}=d_{k}-l_{k}\).
5. Set \(\Phi_{k}(d_{k})=0\) if \(l_{k}>d_{k}\), where \(0\leq d_{k}\leq d\).
6. Calculate \(\Phi_{k}(d_{k})\), the minimum value of RHS of \((11)\) for \(l_{k};\;0\leq l_{k}\leq d_{k}\).
7. Record \(\Phi_{k}(d_{k})\) and \(l_{k}\).
8. For \(k\geq3, ..., L\), go to Step 4.
9. At \(k=L, \; \Phi_{L}(d)\) is obtained and hence the optimum value \(l_{L}^{*}\) of \(l_{L}\) is obtained.
10. At \(k=L-1\), using the backward calculation for \(d_{L-1}=d-l_{L}^{*}\), read the value of \(\Phi_{L-1}(d_{L-1})\) and hence the optimum value \(l_{L-1}^{*}\) of \(l_{L-1}\).
11. Repeat Step \((10)\) until the optimum value \(l_{1}^{*}\) of \(l_{1}\) is obtained from \(\Phi_{1}(d_{1})\).

5 Optimum Sample Sizes Using Neyman Allocation

When OSB \((y_{h}, y_{h-1})\) have been determined, the Optimum Sample Sizes (OSS) \(n_{h}; h=1,2,...,L\,\) that minimizes the variance of the estimate can easily be computed. The sample size \(n_{h}\) are obtained for a fixed total sample of size \(n\) under the Neyman allocation for \(h=1,2,...,L\,\) and given as follows:

\[\begin{equation}\label{Ney_alloc} n_{h} = n\,\frac{W_{h}S_{h}}{\sum_{h=1}^{L}W_{h}S_{h}} \tag{13} \end{equation}\]

where \(W_{h}\) and \(S_{h}\) are derived in terms of the optimum boundary points \((y_{h}, y_{h-1})\).

In Neyman allocation, the total sample size is proportional to the stratum size multiplied by the standard deviation of the stratum. If the variances are specified correctly, Neyman allocation will give an estimator with smaller variance compared to proportional allocation (Lohr (2009)).

In equation (13), it is also worth noting that the OSB \((y_{h}, y_{h-1})\) are so obtained that \(n_{h}\) must satisfy the restrictions:

\[\begin{equation}\label{res} 1\leq n_{h}\leq N_{h}, \tag{14} \end{equation}\]

where \(N_{h}=NW_{h}\). Thus, the restriction (14) indicates that the \(h^ {th}\) stratum must form with at least one unit and also avoid the problem of over-sampling.

6 Overview of Package Functionalities

For the numerical illustrations and application of the package, some of the real datasets such as ‘sugarcane’ of Khan, Reddy, et al. (2015), ‘anaemia’ of Reddy et al. (2014); ‘hies’ and ‘math’ data are provided in the stratifyR. The ‘quakes’ and ‘Boston’ data provided in the datasets package in R are also used for illustration purposes. The stratifyR package is also tested on some published and commonly-used datasets such as ‘UScities’ and ‘UScolleges’ data from Cochran (1961), ‘Debtors’ data of Gunning and Horgan (2004), ‘REV84’ variable for ‘Swedish municipalities’ data from Särdnal et al. (1992) and ‘MRTS’ variable from ‘Statistics Canada Monthly Retail Trade Survey’ together with ‘SHS’ data collected in ‘Statistics Canada Survey of Household Spending’. For those distributions where real data is not found in literature, data is simulated to demonstrate the application of the package in this documentaion.

For a user, there are two different routes available in the package and these are basically dependent on the type of stratification problem to be solved. It could either be a data-based (i.e., when the dataset of the stratification variable is available) or a distribution-based (i.e., when dataset is not available) stratification problem. Whether stratification is based on data or not, the idea is that the problem is formulated as an MPP using the estimated (with available data) or assumed (with unavailable data) distribution of the data set. There are numerous functions created in the package, however, there are only a few major functions that are used by the two different types of problems being studied in univariate stratification.

If it is a data-based problem, the function used is strata.data() and the user has to provide as input arguments: the data, the number of strata (\(L\)) and the fixed sample size (\(n\)). For the distribution-based problem, the function used is strata.distr() and the user has to provide the name of the assumed distribution, number of strata (\(L\)), possible range of data (distance), fixed sample size (\(n\)) and the population size (\(N\)). The following two subsections delve a little deeper into the workings surrounding the two functions: strata.data() and strata.distr().

6.1 The Function strata.data()

This function computes the OSB, OSS, and other important quantities from univariate survey populations by employing the methodology proposed by Khan et al. (2002) Khan et al. (2003), Khan et al. (2005), Khan et al. (2008), Khan et al. (2009), Nand and Khan (2009), Khan, Rao, et al. (2015), Khan, Reddy, et al. (2015), Reddy et al. (2018) and Reddy and Khan (2019). Their method uses the estimated distribution of the data and formulates the problem of determining OSB as a Mathematical Programming Problem which is an optimization problem that is solved by the DP technque. The OSB obtained are optimal solutions that are used to calculate the OSS under Neyman allocation. The usage and arguments are as follows:

    strata.data(data, h, n, cost=FALSE, ch=NULL,
                method=c("dp","cobyla","global"),
                n_starts=20L, max_iter=2000L, tol=1e-9, verbose=FALSE)

    data     - A vector: data containing every unit of the survey population
    h        - A numeric: number of strata to be sampled. The default is 2
    n        - A numeric: fixed total sample size
    cost     - A logical: stratum cost. Default cost=FALSE.
    ch       - A numeric: denotes a vector of stratum costs. Default ch=NULL.
    method   - A character: optimisation method. One of "dp" (default), "cobyla" or "global".
    n_starts - An integer: number of random restarts for COBYLA. Default 20.
    max_iter - An integer: maximum function evaluations per COBYLA run. Default 2000.
    tol      - A numeric: convergence tolerance for COBYLA. Default 1e-9.
    verbose  - A logical: if TRUE, prints per-start diagnostics. Default FALSE.
    

To show the application of the strata.data() function, an example of the command used and its output from the package is given below. The problem uses the ‘mag’ variable from the ‘quakes’ data (with a population of \(N=1000\)) available from the datasets package in R. To construct a 2-strata solution with a fixed sample size of \(n=300\), we use the following codes:

data(quakes)
head(quakes)
#>      lat   long depth mag stations
#> 1 -20.42 181.62   562 4.8       41
#> 2 -20.62 181.03   650 4.2       15
#> 3 -26.00 184.10    42 5.4       43
#> 4 -17.97 181.66   626 4.1       19
#> 5 -20.42 181.96   649 4.0       11
#> 6 -19.68 184.31   195 4.0       12
mag <- quakes$mag
length(mag)
#> [1] 1000
hist(mag) #to see the distribution

# In development, source the 2.0-1 R files to override the installed 1.0-5 functions.
# The library() call above is still needed to make package datasets available.
# When the 2.0-1 package is properly installed, the if-block below is a no-op.
local({
  r_dir <- file.path("..", "R")
  if (dir.exists(r_dir))
    invisible(lapply(list.files(r_dir, pattern = "\\.R$", full.names = TRUE), source))
})
res <- strata.data(mag, h = 2, n=300) # a 2-strata solution
#> Warning in checkparamlist(arg_startfix$start.arg, arg_startfix$fix.arg, : Some
#> parameter names have no starting/fixed value but have a default value: mean.
#> The program is running, it'll take some time!
summary(res)
#> <div style='margin-bottom:0.5rem'><span style='color:#0b7285;font-weight:700'>Stratification summary</span></div>
#> <div><span style='color:#0b7285;font-weight:600'>Strata (h):</span> 2 &nbsp; <span style='color:#0b7285;font-weight:600'>n:</span> 300 &nbsp; <span style='color:#0b7285;font-weight:600'>N:</span> 1000</div>
#> <div><span style='color:#0b7285;font-weight:600'>Data range:</span> [4.0000, 6.4000]  (d = 2.4000)</div>
#> <div><span style='color:#0b7285;font-weight:600'>Best-fit distribution:</span> lnorm</div>
#> 
#> 
#> 
#> Table: Parameter estimate(s)
#> 
#> |Parameter | Estimate|
#> |:---------|--------:|
#> |meanlog   | 1.526810|
#> |sdlog     | 0.085036|

Per-stratum results and totals

Strata	OSB	Wh	Vh	WhSh	nh	Nh	fh
1	4.68	0.58	0.03	0.109	140	585	0.24
2	6.4	0.42	0.09	0.124	160	415	0.38
Total		1.00	0.12	0.233	300	1000	0.30

All calculations have been rounded off to 2 decimal places, hence, the individual stratum solutions provided in the tables may not always add up to the totals.

6.2 The Function strata.distr()

This function is also used to compute the OSB, OSS, and other important quantities from univariate survey populations by employing the methodology proposed by Khan et al given earlier. Its algorithm is quite similar to that of the strata.data(), however, its functionality is applied to the case where the dataset of the population is not available and the distributonal assumptions of the study variable are strictly required. Another caveat for such distribution-based stratification is that the distr.alloc() function uses the probability density functions of the assumed distributions and integration rules presented by equations (4)-(6) to calculate the stratum sample sizes. It must be noted that this function works on ideal distributions that assumes the parameters chosen by the user. The usage and arguments are as follows:

    strata.distr(h, initval = NULL, dist = NULL,
                 distr = c("pareto", "triangle", "rtriangle", "weibull", "gamma",
                 "exp", "unif","norm", "lnorm", "cauchy"), params = c(shape=0,
                 scale=0, rate=0, gamma=0, location=0, mean=0, sd=0, meanlog=0,
                 sdlog=0, min=0, max=0, mode=0), n, N, cost=FALSE, ch=NULL,
                 method=c("dp","cobyla","global"),
                 n_starts=20L, max_iter=2000L, tol=1e-9, verbose=FALSE)

    h        - A numeric: number of strata to be sampled
    initval  - A numeric: initial value of the assumed distribution
    dist     - A numeric: distance or range of the assumed distribution
    distr    - A character: the assumed distribution of the hypothetical population
    params   - A list: parameters of the assumed distribution
    n        - A numeric: fixed total sample size
    N        - A numeric: fixed population size
    cost     - A logical: stratum cost. Default cost=FALSE.
    ch       - A numeric: denotes a vector of stratum costs. Default ch=NULL.
    method   - A character: optimisation method. One of "dp" (default), "cobyla" or "global".
    n_starts - An integer: number of random restarts for COBYLA. Default 20.
    max_iter - An integer: maximum function evaluations per COBYLA run. Default 2000.
    tol      - A numeric: convergence tolerance for COBYLA. Default 1e-9.
    verbose  - A logical: if TRUE, prints per-start diagnostics. Default FALSE.
    

The algorithm for strata.distr() is quite similar to the strata.data() for the contruction of OSB. It is only at the sample allocation (OSS) stage that the two are different. strata.distr() is the main function and once the OSB have been computed, this function uses the distr.alloc() function which uses the numerical integration rules for different distibutions (i.e., the equations (4)-(6)) to compute the OSS.

To show the application of the strata.distr() function, let us construct a 2-strata solution assuming that the dataset of the stratification variable is not available but its distribution and estimated parameters are. Let us consider the ‘depth’ variable from the ‘quakes’ dataset from the datasets package, which has a Triangular distribution with parameters \(min=39.99998, max=680, mode=39.99999\). It starts at an initial value of \(40\) and has a distance (range) of \(640\) with a fixed sample size of \(n=300\) from a population of \(N=1000\) seismic events. Thus, we use the following commands:

data(quakes) 
depth <- quakes$depth
hist(depth) #see distribution

min(depth); max(depth); d=max(depth)-min(depth);d #min, max and range of data 
#> [1] 40
#> [1] 680
#> [1] 640
# the 2-strata solution is
res <- strata.distr(h=2, initval=40, dist=640, distr = "triangle",
             params = c(min=39.99998, max=680, mode=39.99999), n=300, N=1000)
#> The program is running, it'll take some time!
summary(res)
#> <div style='margin-bottom:0.5rem'><span style='color:#0b7285;font-weight:700'>Stratification summary</span></div>
#> <div><span style='color:#0b7285;font-weight:600'>Strata (h):</span> 2 &nbsp; <span style='color:#0b7285;font-weight:600'>n:</span> 300 &nbsp; <span style='color:#0b7285;font-weight:600'>N:</span> 1000</div>
#> <div><span style='color:#0b7285;font-weight:600'>Data range:</span> [40.0000, 680.0000]  (d = 640.0000)</div>
#> <div><span style='color:#0b7285;font-weight:600'>Best-fit distribution:</span> triangle</div>
#> 
#> 
#> 
#> Table: Parameter estimate(s)
#> 
#> |Parameter |  Estimate|
#> |:---------|---------:|
#> |min       |  39.99998|
#> |max       | 680.00000|
#> |mode      |  39.99999|

Per-stratum results and totals

Strata	OSB	Wh	Vh	WhSh	nh	Nh	fh
1	266.72	0.58	4217.34	37.861	145	583	0.25
2	680	0.42	9488.91	40.620	155	417	0.37
Total		1.00	13706.25	78.481	300	1000	0.30

7 Alternative Optimisation Methods: COBYLA and GLOBAL

7.1 Motivation

The Dynamic Programming solver described in Section 4 is the foundation of the original stratifyR package and guarantees finding the globally optimum stratum boundaries on a discrete grid of 1,000 candidate positions. While this is accurate and reliable, the exhaustive nature of the grid search means computation time grows with the number of strata \(h\) and the resolution of the grid. For applications involving a large number of strata, or when fast approximate results are needed, for instance during exploratory analysis, simulation studies, or when stratifying a large number of variables simultaneously, it is advantageous to employ a faster optimisation method.

stratifyR 2.0-1 introduces two alternative optimisation methods, accessible through the method argument in both strata.data() and strata.distr():

• method = "cobyla": a multi-start gradient-free local optimiser based on COBYLA
• method = "global": a two-phase global optimiser combining DIRECT-L with COBYLA refinement

The original DP solver remains available as method = "dp" and is the default for both functions, ensuring full backward compatibility with version 1.0-5. The two new methods are intended as complements to DP, trading some accuracy for speed, and are most effective when applied to distribution-based stratification (strata.distr) where the objective function is smooth and continuous.

7.2 The COBYLA Method (method = "cobyla")

COBYLA (Constrained Optimisation By Linear Approximations) is a gradient-free, derivative-free nonlinear optimisation algorithm developed by Powell (1994). It minimises a nonlinear objective function subject to nonlinear inequality constraints by constructing successive linear approximations to both the objective and constraint functions. Unlike gradient-based methods, COBYLA does not require the objective function to be differentiable, making it in principle applicable to the stratification objective \(\sum_{h=1}^{L} W_h S_h\).

In stratifyR, COBYLA is applied with multiple random starting points (controlled by the n_starts argument, default 20) to reduce sensitivity to local minima. At each start, it searches for the \((h-1)\) interior boundary positions that minimise the objective, subject to constraints that keep boundaries ordered and prevent any stratum from collapsing to near-zero width. The best solution found across all starts is returned. If the solution is detected as degenerate (a stratum has collapsed), the solver automatically falls back to the DP method.

For strata.distr, COBYLA optimises the same analytical objective function evaluated by the DP solver, but treats the boundary positions as continuous variables rather than a discrete grid. Because this function is smooth and differentiable almost everywhere, COBYLA can navigate it effectively and typically converges to a solution close to the DP optimum in a fraction of the computation time.

For strata.data, COBYLA optimises an empirical objective computed directly from the raw data (stratum weights \(W_h\) and standard deviations \(S_h\) calculated by binning actual observations at each candidate boundary). This objective is piecewise constant, it only changes value when a boundary crosses an actual data point, which means continuous optimisers cannot reliably find the global minimum. For this reason, method = "cobyla" and method = "global" are most suitable for strata.distr, and method = "dp" is always recommended for strata.data.

Advantages of COBYLA:

• Substantially faster than DP, particularly for larger \(h\), making it well-suited to exploratory work and simulation studies
• No derivative information required; the algorithm is self-contained
• Multiple random restarts reduce (but do not eliminate) sensitivity to local minima
• Well-suited to distribution-based stratification where the analytical objective is smooth and continuous

Disadvantages of COBYLA:

• Not guaranteed to find the global optimum; the result may vary slightly between runs due to random starting points
• For data-based stratification, cannot reliably match the DP result because the empirical Wh·Sh objective is piecewise constant
• Degeneracy risk (collapsed strata) is handled automatically by a fallback to DP, but this removes the speed advantage in such cases

7.3 The GLOBAL Method (method = "global")

The "global" method employs a two-phase approach designed to improve upon single-phase COBYLA by systematically exploring the search space before local refinement.

Phase 1 — DIRECT-L global search. The DIRECT-L algorithm (Dividing RECTangles, locally-biased variant; Gablonsky and Kelley (2001)) partitions the feasible search space into hyper-rectangles and evaluates the objective at the centre of each. It then subdivides rectangles that are either (a) small and promising (local exploitation), or (b) large and unexplored (global coverage), ensuring that the entire feasible region is investigated over time. Unlike COBYLA, DIRECT-L does not depend on random starting points and provides deterministic, reproducible exploration of the search space. The algorithm continues until the evaluation budget (governed by max_iter) is exhausted.

Phase 2 — COBYLA local refinement. The best boundary position found by DIRECT-L in Phase 1 is used as the starting point for a COBYLA local refinement step, which polishes the solution to the required tolerance. This combination leverages the global coverage of DIRECT-L and the precise local convergence of COBYLA.

For strata.distr, this two-phase approach is most effective: DIRECT-L identifies the basin of attraction containing the global minimum of the smooth analytical objective, and COBYLA precisely locates the minimum within that basin. The result typically agrees very closely with the DP optimum while running considerably faster for larger \(h\).

As with COBYLA, if the resulting solution is detected as degenerate, an automatic fallback to DP is applied.

Advantages of the GLOBAL method:

• Deterministic global search (Phase 1) avoids sensitivity to random starting points
• More thorough coverage of the feasible space than multi-start COBYLA
• Closest among the three methods to matching DP results for strata.distr
• Well-suited to distribution-based stratification problems where the highest accuracy and speed are both desired

Disadvantages of the GLOBAL method:

• Slower than COBYLA (though still faster than DP for large \(h\)) due to DIRECT-L’s evaluation budget
• For data-based stratification, does not reliably match DP, for the same reasons as COBYLA
• Degeneracy fallback to DP is applied automatically if the global search returns collapsed strata

7.4 Summary: When to Use Each Method

The choice of method depends on whether raw data are available and what the priorities are in terms of speed versus accuracy.

For data-based stratification (strata.data), the DP solver is the recommended choice in all situations. The empirical \(W_h S_h\) objective is piecewise constant, it only changes when a boundary crosses a data point and is flat everywhere else, which prevents continuous optimisers from reliably navigating to the global minimum. The DP solver, being an exhaustive discrete grid search, is unaffected by this and always returns the globally optimal grid solution.

For distribution-based stratification (strata.distr), all three methods optimise the same smooth, differentiable analytical objective, and the choice is a trade-off between speed and closeness to the DP optimum. COBYLA is the fastest option and is well-suited to exploratory or iterative work. GLOBAL is slower than COBYLA but typically agrees more closely with DP, making it preferable when the highest quality approximate result is needed without incurring the full cost of DP. DP remains the gold standard for final, publication-quality results.

In practice, a sensible workflow for distribution-based problems is: use method = "cobyla" or method = "global" to quickly explore a range of \(h\) values or parameter settings, then confirm the chosen configuration with method = "dp".

7.5 Application: Comparing Methods for strata.distr

The following example uses the Weibull distribution fitted to the Iron variable of the ‘anaemia’ data to compare the three methods for a 3-strata solution. For distribution-based stratification, all three methods should produce very similar OSB and WhShTot values because the analytical objective is smooth.

# Iron variable from the anaemia data follows Weibull(shape=2.1446, scale=13.7907)
# as established in Section 7.4. Using those known parameters directly here.
data(anaemia)
Iron <- anaemia$Iron

# DP (reference solution)
res_dp_d <- strata.distr(h=3, initval=1.5, dist=33.2,
              distr="weibull", params=c(shape=2.1446, scale=13.7907),
              n=350, N=724, method="dp")
#> The program is running, it'll take some time!
summary(res_dp_d)
#> <div style='margin-bottom:0.5rem'><span style='color:#0b7285;font-weight:700'>Stratification summary</span></div>
#> <div><span style='color:#0b7285;font-weight:600'>Strata (h):</span> 3 &nbsp; <span style='color:#0b7285;font-weight:600'>n:</span> 350 &nbsp; <span style='color:#0b7285;font-weight:600'>N:</span> 724</div>
#> <div><span style='color:#0b7285;font-weight:600'>Data range:</span> [1.5000, 34.7000]  (d = 33.2000)</div>
#> <div><span style='color:#0b7285;font-weight:600'>Best-fit distribution:</span> weibull</div>
#> 
#> 
#> 
#> Table: Parameter estimate(s)
#> 
#> |Parameter | Estimate|
#> |:---------|--------:|
#> |shape     |   2.1446|
#> |scale     |  13.7907|

Per-stratum results and totals

Strata	OSB	Wh	Vh	WhSh	nh	Nh	fh
1	9.66	0.37	4.53	0.775	116	270	0.43
2	16.31	0.39	3.56	0.733	110	282	0.39
3	34.7	0.24	11.94	0.822	123	173	0.71
Total		1.00	20.02	2.330	350	724	0.48

# COBYLA (fast approximation)
res_cob_d <- strata.distr(h=3, initval=1.5, dist=33.2,
               distr="weibull", params=c(shape=2.1446, scale=13.7907),
               n=350, N=724, method="cobyla")
summary(res_cob_d)
#> <div style='margin-bottom:0.5rem'><span style='color:#0b7285;font-weight:700'>Stratification summary</span></div>
#> <div><span style='color:#0b7285;font-weight:600'>Strata (h):</span> 3 &nbsp; <span style='color:#0b7285;font-weight:600'>n:</span> 350 &nbsp; <span style='color:#0b7285;font-weight:600'>N:</span> 724</div>
#> <div><span style='color:#0b7285;font-weight:600'>Data range:</span> [1.5000, 34.7000]  (d = 33.2000)</div>
#> <div><span style='color:#0b7285;font-weight:600'>Best-fit distribution:</span> weibull</div>
#> 
#> 
#> 
#> Table: Parameter estimate(s)
#> 
#> |Parameter | Estimate|
#> |:---------|--------:|
#> |shape     |   2.1446|
#> |scale     |  13.7907|

Per-stratum results and totals

Strata	OSB	Wh	Vh	WhSh	nh	Nh	fh
1	9.66	0.37	4.53	0.775	116	270	0.43
2	16.3069	0.39	3.55	0.733	110	281	0.39
3	34.7	0.24	11.94	0.822	124	173	0.71
Total		1.00	20.02	2.330	350	724	0.48

# GLOBAL (DIRECT-L + COBYLA)
res_glob_d <- strata.distr(h=3, initval=1.5, dist=33.2,
                distr="weibull", params=c(shape=2.1446, scale=13.7907),
                n=350, N=724, method="global")
summary(res_glob_d)
#> <div style='margin-bottom:0.5rem'><span style='color:#0b7285;font-weight:700'>Stratification summary</span></div>
#> <div><span style='color:#0b7285;font-weight:600'>Strata (h):</span> 3 &nbsp; <span style='color:#0b7285;font-weight:600'>n:</span> 350 &nbsp; <span style='color:#0b7285;font-weight:600'>N:</span> 724</div>
#> <div><span style='color:#0b7285;font-weight:600'>Data range:</span> [1.5000, 34.7000]  (d = 33.2000)</div>
#> <div><span style='color:#0b7285;font-weight:600'>Best-fit distribution:</span> weibull</div>
#> 
#> 
#> 
#> Table: Parameter estimate(s)
#> 
#> |Parameter | Estimate|
#> |:---------|--------:|
#> |shape     |   2.1446|
#> |scale     |  13.7907|

Per-stratum results and totals

Strata	OSB	Wh	Vh	WhSh	nh	Nh	fh
1	9.66	0.37	4.53	0.775	116	270	0.43
2	16.3069	0.39	3.55	0.733	110	281	0.39
3	34.7	0.24	11.94	0.822	124	173	0.71
Total		1.00	20.02	2.330	350	724	0.48

# Direct comparison of all three methods
cat("DP     OSB:", round(res_dp_d$OSB, 2),   "  WhShTot:", round(res_dp_d$WhShTot, 4), "\n")
#> DP     OSB: 9.66 16.31 34.7   WhShTot: 2.33
cat("COBYLA OSB:", round(res_cob_d$OSB, 2),  "  WhShTot:", round(res_cob_d$WhShTot, 4), "\n")
#> COBYLA OSB: 9.66 16.31   WhShTot: 2.33
cat("GLOBAL OSB:", round(res_glob_d$OSB, 2), "  WhShTot:", round(res_glob_d$WhShTot, 4), "\n")
#> GLOBAL OSB: 9.66 16.31   WhShTot: 2.33

For distribution-based stratification, DP and GLOBAL typically agree very closely, since both locate the global minimum of the same smooth analytical objective — DP via exhaustive discrete grid search, and GLOBAL via deterministic DIRECT-L exploration followed by local refinement. COBYLA also agrees well for most distributions, but on objective landscapes with a pronounced local minimum — such as Weibull with certain shape/scale combinations — standalone COBYLA may converge to a suboptimal solution rather than the global minimum, because its linear-approximation local search is sensitive to the basin of attraction from which it starts. In such cases method = "global" is preferred over method = "cobyla" for distribution-based stratification.

7.6 Application: Comparing Methods for strata.data

The following example uses the Iron variable directly to compare the three methods for a 2-strata data-based solution. This illustrates the fundamental limitation of continuous optimisers on empirical data.

data(anaemia)
Iron <- anaemia$Iron
Iron[Iron == min(Iron)] <- -0.001  # as per standard package convention

# DP (recommended for strata.data)
res_dp_s <- strata.data(Iron, h=2, n=350, method="dp")
#> Warning in checkparamlist(arg_startfix$start.arg, arg_startfix$fix.arg, : Some
#> parameter names have no starting/fixed value but have a default value: mean.
#> The program is running, it'll take some time!
summary(res_dp_s)
#> <div style='margin-bottom:0.5rem'><span style='color:#0b7285;font-weight:700'>Stratification summary</span></div>
#> <div><span style='color:#0b7285;font-weight:600'>Strata (h):</span> 2 &nbsp; <span style='color:#0b7285;font-weight:600'>n:</span> 350 &nbsp; <span style='color:#0b7285;font-weight:600'>N:</span> 724</div>
#> <div><span style='color:#0b7285;font-weight:600'>Data range:</span> [-0.0010, 34.7000]  (d = 34.7010)</div>
#> <div><span style='color:#0b7285;font-weight:600'>Best-fit distribution:</span> norm</div>
#> 
#> 
#> 
#> Table: Parameter estimate(s)
#> 
#> |Parameter |  Estimate|
#> |:---------|---------:|
#> |mean      | 12.194472|
#> |sd        |  6.026547|

Per-stratum results and totals

Strata	OSB	Wh	Vh	WhSh	nh	Nh	fh
1	12.58	0.58	8.83	1.724	171	420	0.41
2	34.7	0.42	18.49	1.805	179	304	0.59
Total		1.00	27.32	3.530	350	724	0.48

# COBYLA
res_cob_s <- strata.data(Iron, h=2, n=350, method="cobyla")
#> Warning in checkparamlist(arg_startfix$start.arg, arg_startfix$fix.arg, : Some
#> parameter names have no starting/fixed value but have a default value: mean.
summary(res_cob_s)
#> <div style='margin-bottom:0.5rem'><span style='color:#0b7285;font-weight:700'>Stratification summary</span></div>
#> <div><span style='color:#0b7285;font-weight:600'>Strata (h):</span> 2 &nbsp; <span style='color:#0b7285;font-weight:600'>n:</span> 350 &nbsp; <span style='color:#0b7285;font-weight:600'>N:</span> 724</div>
#> <div><span style='color:#0b7285;font-weight:600'>Data range:</span> [-0.0010, 34.7000]  (d = 34.7010)</div>
#> <div><span style='color:#0b7285;font-weight:600'>Best-fit distribution:</span> norm</div>
#> 
#> 
#> 
#> Table: Parameter estimate(s)
#> 
#> |Parameter |  Estimate|
#> |:---------|---------:|
#> |mean      | 12.194472|
#> |sd        |  6.026547|

Per-stratum results and totals

Strata	OSB	Wh	Vh	WhSh	nh	Nh	fh
1	13.2509	0.62	9.74	1.948	194	452	0.43
2	34.7	0.38	17.35	1.565	156	272	0.57
Total		1.00	27.09	3.513	350	724	0.48

# GLOBAL
res_glob_s <- strata.data(Iron, h=2, n=350, method="global")
#> Warning in checkparamlist(arg_startfix$start.arg, arg_startfix$fix.arg, : Some
#> parameter names have no starting/fixed value but have a default value: mean.
summary(res_glob_s)
#> <div style='margin-bottom:0.5rem'><span style='color:#0b7285;font-weight:700'>Stratification summary</span></div>
#> <div><span style='color:#0b7285;font-weight:600'>Strata (h):</span> 2 &nbsp; <span style='color:#0b7285;font-weight:600'>n:</span> 350 &nbsp; <span style='color:#0b7285;font-weight:600'>N:</span> 724</div>
#> <div><span style='color:#0b7285;font-weight:600'>Data range:</span> [-0.0010, 34.7000]  (d = 34.7010)</div>
#> <div><span style='color:#0b7285;font-weight:600'>Best-fit distribution:</span> norm</div>
#> 
#> 
#> 
#> Table: Parameter estimate(s)
#> 
#> |Parameter |  Estimate|
#> |:---------|---------:|
#> |mean      | 12.194472|
#> |sd        |  6.026547|

Per-stratum results and totals

Strata	OSB	Wh	Vh	WhSh	nh	Nh	fh
1	13.2558	0.62	9.74	1.948	194	452	0.43
2	34.7	0.38	17.35	1.565	156	272	0.57
Total		1.00	27.09	3.513	350	724	0.48

# Direct comparison
cat("DP     OSB:", round(res_dp_s$OSB, 2),   "  WhShTot:", round(res_dp_s$WhShTot, 4), "\n")
#> DP     OSB: 12.58 34.7   WhShTot: 3.53
cat("COBYLA OSB:", round(res_cob_s$OSB, 2),  "  WhShTot:", round(res_cob_s$WhShTot, 4), "\n")
#> COBYLA OSB: 13.25   WhShTot: 3.513
cat("GLOBAL OSB:", round(res_glob_s$OSB, 2), "  WhShTot:", round(res_glob_s$WhShTot, 4), "\n")
#> GLOBAL OSB: 13.26   WhShTot: 3.513

The results show that for data-based stratification, COBYLA and GLOBAL may report a lower WhShTot than DP, with different OSB. This is not because DP is performing worse — it is because the two approaches are minimising different objectives. DP optimises a theoretical \(W_h S_h\) computed analytically under the fitted distribution (via data.root), while COBYLA and GLOBAL directly minimise the empirical \(W_h S_h\) computed from the actual data counts and standard deviations. The reported WhShTot is always empirical, so methods that directly minimise the empirical objective will naturally tend to report a lower value. However, the DP boundaries are theoretically justified: they minimise the expected Neyman cost under the best-fit distributional model, in the same way as strata.distr. The empirical boundaries found by COBYLA and GLOBAL are tailored to the specific sample and do not generalise to the underlying population in the same principled way. For this reason, method = "dp" remains the recommended and default choice for strata.data.

8 Application to Numerous Survey Populations

As discussed earlier, the stratifyR package is able to handle ten continuous distributions that are quite commonly-used in real-life situations. This section presents a brief overview of these distributions and the application of the proposed method of stratification using real or simulated data which follows a particular distribution. Examples for hypothetical distributions are also presented. For the sake of brevity, the mathematical formulations of the problem of determining the OSB and the DP solution procedure are presented only for the Pareto Type II variable. For all other distributions, only the examples are presented to illustrate the application of the package.

8.1 Stratification for a Survey Variable with Pareto Type II Distribution

Let the study variable follow the Pareto Type II distribution on the domain of [\(0, +\infty\)], its two-parameter probability density function with a state space \(y\geq 0\) is given by: \[\begin{equation} f(y; s,a)=\dfrac{a\,s^{a}}{(y+s)^{a+1}}, \;\;\;\;\;a,s > 0 \tag{15} \end{equation}\]

where \(\alpha > 0\) is the shape parameter and \(s>0\) is the scale parameter of the distribution.

8.1.1 MPP Formulation for Pareto Type II Distribution

If the study variable \(y\) follows Pareto II (or Lomax) distribution (i.e., \(y\sim P(a, s)\)) with density function given by (15). By using equations (4)-(6), the formulated MPP given in (10) could be expressed as: \[\begin{eqnarray} \textrm{Minimize} \;\;\; \sum_{h=1}^{L} \textit{SQRT} && \Biggl\{as^{2a} \left[\dfrac{(y_{h-1}+l_{h}+s)^{a}-(y_{h-1}+s)^{a}}{(y_{h-1}+s)^{a}(y_{h-1}+l_{h}+s)^{a}}\right] \nonumber \\[10pt] && \times\left[\dfrac{(y_{h-1}+l_{h}+s)^{2-a}}{2-a} - \dfrac{2s(y_{h-1}+l_{h}+s)^{1-a}}{1-a} \right. \nonumber\\[10pt] && - \dfrac{s^2(y_{h-1}+l_{h}+s)^{-a}}{a} -\dfrac{(y_{h-1}+s)^{2-a}}{2-a}\nonumber\\[10pt] && + \left. \dfrac{2s(y_{h-1}+s)^{1-a}}{1-a} + \dfrac{s^2(y_{h-1}+s)^{-a}}{a}\right] \nonumber\\[10pt] && \times \dfrac{s^{2a}}{(1-a)^{2}}\left[\dfrac{a(y_{h-1}+l_{h})+s}{(y_{h-1}+l_{h}+s)^{a}} - \dfrac{ay_{h-1}+s}{(y_{h-1}+s)^{a}} \right]^{2}\Biggr\}\nonumber\\[15pt] \textrm{subject to} && \;\;\;\; \sum_{h=1}^{L}l_{h} = d,\nonumber \\ \textrm{and} && \;\;\;\; l_{h} \geq 0; \;\; h = 1,2,...,L. \tag{16} \end{eqnarray}\]

where \(d=y_{L}-y_{0}\), \(a\) and \(s\) are parameters of the Pareto Type II distribution.

8.1.2 DP Solution for Pareto Type II Distribution

To solve the MPP (16) using the DP technique as a solution procedure, we apply the algorithm, that is, the solution proceure using Dynamic Progrmming technique discussed earlier in Section 4. After substituting the quatity \(y_{h-1}=y_{0} + d_{h} - l_{h}\), the recurrence relations (11) and (12) are reduced to:

For the first stage, \(k=1\), at \(l_{1}^{*}=d_{1}\):

\[\begin{eqnarray} \Phi_{1}d_{1}&=& \textrm{Sqrt}\Biggl\{ as^{2a}\left[\dfrac{(d_{1}+y_{0}+s)^{a}-(y_{0}+s)^{a}}{(y_{0}+s)^{a}(d_{1}+y_{0}+s)^{a}}\right] \nonumber\\[10pt] &&\times\left[\dfrac{(d_{1}+y_{0}+s)^{2-a}}{2-a} - \dfrac{2s(d_{1}+y_{0}+s)^{1-a}}{1-a} \right. \nonumber\\[10pt] &&- \dfrac{s^2(d_{1}+y_{0}+s)^{-a}}{a} -\dfrac{(y_{0}+s)^{2-a}}{2-a}\nonumber\\[10pt] &&+\left. \dfrac{2s(y_{0}+s)^{1-a}}{1-a} + \dfrac{s^2(y_{0}+s)^{-a}}{a}\right] \nonumber\\[10pt] &&\times \dfrac{s^{2a}}{(1-a)^{2}}\left[\dfrac{a(d_{1}+y_{0})+s}{(d_{1}+y_{0}+s)^{a}} - \dfrac{ay_{0}+s}{(y_{0}+s)^{a}} \right]^{2}\Biggr\} \tag{17} \end{eqnarray}\]

And for the stages \(k\geq2\):

\[\begin{eqnarray} \Phi_{k}d_{k}&=&\min_{0\leq l_{k}\leq d_{k}}\Biggl\{ \textrm{Sqrt}\Biggl\{ as^{2a}\left[\dfrac{(d_{k}+y_{0}+s)^{a}-(d_{k}+l_{k}+y_{0}+s)^{a}}{(d_{k}+l_{k}+y_{0}+s)^{a}(d_{k}+y_{0}+s)^{a}}\right] \nonumber\\[10pt] &&\times\left[\dfrac{(d_{k}+y_{0}+s)^{2-a}}{2-a} - \dfrac{2s(d_{k}+y_{0}+s)^{1-a}}{1-a} \right. \nonumber\\[10pt] &&- \dfrac{s^2(d_{k}+y_{0}+s)^{-a}}{a} -\dfrac{(d_{k}+l_{k}+y_{0}+s)^{2-a}}{2-a}\nonumber\\[10pt] &&+\left. \dfrac{2s(d_{k}+l_{k}+y_{0}+s)^{1-a}}{1-a} + \dfrac{s^2(d_{k}+l_{k}+y_{0}+s)^{-a}}{a}\right] \nonumber\\[10pt] &&\times \dfrac{s^{2a}}{(1-a)^{2}}\left[\dfrac{a(d_{k}+y_{0})+s}{(d_{k}+y_{0}+s)^{a}}- \dfrac{a(d_{k}+l_{k}+y_{0})+s}{(d_{k}+l_{k}+y_{0}+s)^{a}} \right]^{2} \Biggr\} + \Phi_{k-1}(d_{k}-l_{k})\Biggr\}. \tag{18} \end{eqnarray}\]

The recurrence relations (17) and (18) are solved using the DP technique to determine the OSB.

8.1.3 A Numerical Example for Pareto Type II Distribution

A dataset for a univariate population of size \(N=5000\) with the study variable that follows Pareto Type II distribution (\(pareto\_data\)) was simulated using parameters \(shape=5\) and \(scale=8\) to demonstrate the application of the strata.data() function to determine the OSB and other quantites. The data exhibits a 2-parameter Pareto Type II distribution with the MLE estimates of the parameters as \(shape=5.026907\) and \(scale=8.191676\). The minimum and maximum values in the simulated data are \([y_{0}, y_{L}] = [0.0002193, 38.56871]\), which implies that \(d=38.56849\).

To construct the OSB (a 2-strata solution, i.e., \(h = 2\)) for the \(pareto\_data\) with a fixed total sample size of \(500\), we use the following codes:

set.seed(8235411)
pareto_data <- actuar::rpareto(5000, shape=5, scale=8)

dpareto <- actuar::dpareto
ppareto <- actuar::ppareto
qpareto <- actuar::qpareto

head(pareto_data)
#> [1] 1.8464786 0.9226109 1.1079785 9.7699864 4.5600912 0.0441124
hist(pareto_data, breaks=100)

min(pareto_data); max(pareto_data); d=max(pareto_data)-min(pareto_data);d
#> [1] 0.0002192696
#> [1] 38.56871
#> [1] 38.56849
fit <- fitdistrplus::fitdist(pareto_data, "pareto", start = list(shape = 1, scale = 500))
fit
#> Fitting of the distribution ' pareto ' by maximum likelihood 
#> Parameters:
#>       estimate Std. Error
#> shape 5.026907  0.4337687
#> scale 8.191676  0.8358576
res <- strata.data(pareto_data, h = 2, n=500) # a 2-strata solution
#> Warning in checkparamlist(arg_startfix$start.arg, arg_startfix$fix.arg, : Some
#> parameter names have no starting/fixed value but have a default value: mean.
#> The program is running, it'll take some time!
summary(res)
#> <div style='margin-bottom:0.5rem'><span style='color:#0b7285;font-weight:700'>Stratification summary</span></div>
#> <div><span style='color:#0b7285;font-weight:600'>Strata (h):</span> 2 &nbsp; <span style='color:#0b7285;font-weight:600'>n:</span> 500 &nbsp; <span style='color:#0b7285;font-weight:600'>N:</span> 5000</div>
#> <div><span style='color:#0b7285;font-weight:600'>Data range:</span> [0.0002, 38.5687]  (d = 38.5685)</div>
#> <div><span style='color:#0b7285;font-weight:600'>Best-fit distribution:</span> weibull</div>
#> 
#> 
#> 
#> Table: Parameter estimate(s)
#> 
#> |Parameter | Estimate|
#> |:---------|--------:|
#> |shape     | 0.875786|
#> |scale     | 1.895872|

Per-stratum results and totals

Strata	OSB	Wh	Vh	WhSh	nh	Nh	fh
1	2.73	0.76	0.54	0.560	205	3803	0.05
2	38.57	0.24	11.34	0.806	295	1197	0.25
Total		1.00	11.88	1.366	500	5000	0.10

Similarly, in order to find the OSB and other quantities, we can apply the strata.distr() function to a Pareto Type II population. Let us assume from past knowledge that the study variable in the population follows Pareto Type II distribution with the given attributes such as the \(shape=5.05, scale=8.20, initial\,value=0.15\) and \(distance=38.55\). Then, if a sample of size \(n=500\) is drawn from the population of size \(N=5000\), we can execute the following command to obtain the results:

res <- strata.distr(h=2, initval=0.15, dist=38.55, distr = "pareto",
             params = c(shape=5.05, scale=8.20), n=500, N=5000)
#> The program is running, it'll take some time!
summary(res)
#> <div style='margin-bottom:0.5rem'><span style='color:#0b7285;font-weight:700'>Stratification summary</span></div>
#> <div><span style='color:#0b7285;font-weight:600'>Strata (h):</span> 2 &nbsp; <span style='color:#0b7285;font-weight:600'>n:</span> 500 &nbsp; <span style='color:#0b7285;font-weight:600'>N:</span> 5000</div>
#> <div><span style='color:#0b7285;font-weight:600'>Data range:</span> [0.1500, 38.7000]  (d = 38.5500)</div>
#> <div><span style='color:#0b7285;font-weight:600'>Best-fit distribution:</span> pareto</div>
#> 
#> 
#> 
#> Table: Parameter estimate(s)
#> 
#> |Parameter | Estimate|
#> |:---------|--------:|
#> |shape     |     5.05|
#> |scale     |     8.20|

Per-stratum results and totals

Strata	OSB	Wh	Vh	WhSh	nh	Nh	fh
1	3.21	0.81	0.67	0.592	241	4057	0.06
2	38.7	0.19	11.42	0.637	259	943	0.27
Total		1.00	12.09	1.229	500	5000	0.10

8.2 Stratification for a Survey Variable with Triangular Distribution

Let the study variable follow Triangular distribution on the domain of [\(a, b\)], its three-parameter probability density function with a state space \(y\geq 0\) is given by: \[\begin{equation} f(y) = \begin{cases} \dfrac{2(y-a)}{(b-a)(c-a)}; & y \in [a, c]\\[10pt] \dfrac{2(b-y)}{(b-a)(b-c)}; & y \in (c,b] \\ \end{cases} \tag{19} \end{equation}\]

where \(a\) is the location parameter, \(b\) is the scale parameter and \(c\) is the shape parameter of the distribution.

Then, formulating the problem as an MPP and solving the recurrence relations as discussed above for Pareto II variate, the OSB are obtained.

8.2.1 DP Solution for Triangular Distribution

To solve the MPP formulated for Triangular distribution (19), we apply the algorithm using the DP technique discussed in Section 4. The recurrence relations used to determine the OSB are given by:

For the first stage, \(k=1\), at \(l_{1}^{*}=d_{1}\):

\[\begin{eqnarray} \Phi_{1}d_{1}&=& \textrm{Sqrt}\Biggl\{ \sum_{h=1}^{\lambda_{1}} \dfrac{d_{1}^{2}\sqrt{d_{1}^{2}+6(y_{0}-a)d_{1}+6(y_{0}-a)^{2}}}{3\sqrt{2}(b-a)(c-a)} + \sum_{h=\lambda_{2}}^{L}\dfrac{d_{1}^{2}\sqrt{6(b-y_{0})^{2}-6(b-y_{0})d_{1}+d_{1}^{2}}}{3\sqrt{2}(b-a)(b-c)}\Biggr\} \tag{20} \end{eqnarray}\]

And for the stages \(k\geq2\):

\[\begin{eqnarray} \Phi_{k}d_{k}&=&\min_{0\leq l_{k}\leq d_{k}}\Biggl\{ \textrm{Sqrt}\Biggl\{\sum_{h=1}^{\lambda_{1}} \dfrac{l_{k}^{2}\sqrt{l_{k}^{2}+6a_{k}l_{k}+6a_{k}^{2}}}{3\sqrt{2}(b-a)(c-a)} +\sum_{h=\lambda_{2}}^{L}\dfrac{l_{k}^{2}\sqrt{6b_{k}^{2}-6b_{k}l_{k}+l_{k}^{2}}}{3\sqrt{2}(b-a)(b-c)}\Biggr\} + \Phi_{k-1}(d_{k}-l_{k})\Biggr\} \tag{21} \end{eqnarray}\]

where \(a_{k}=d_{k}-l_{k}+y_{0}-a\) and \(b_{k}=b-d_{k}+l_{k}-y_{0}+a\).

Substituting the values of \(a\), \(b\), \(c\), \(y_{0}\) and \(d\), the OSW (\(l_{h}^{*}\)) and the OSB (\(y_{h}^{*}=y_{h-1}^{*}-l_{h}^{*}\)) are obtained by executing the strata.data() function.

8.2.2 A Numerical Example for Triangular Distribution

Data on Mathematics marks of first year students in a University in Fiji, with a size of \(N=354\) called ‘math’ data is used to demonstrate the application of the stratifyR package on a Triangular population. In this example, the variable `final_marks’ is used - it exhibits a 3-parameter Triangular distribution. The minimum and maximum values in the ‘math’ data are \([y_{0}, y_{L}] = [7, 97]\), which implies that \(d=90\). Fitting a Triangular distribution by fixing \(\min\) and \(\max\) to the sample bounds and estimating only the mode by MLE gives the parameters: \(min = 7\), \(max = 97\) and \(mode \approx 54\).

To construct the OSB (\(h = 2\)) for the `final_marks’ data with a fixed total sample size of \(150\), we use the following code:

data(math)
final_marks <- math$final_marks
hist(final_marks)

res <- strata.data(final_marks, h = 2, n=150) # a 2-strata solution
#> Warning in checkparamlist(arg_startfix$start.arg, arg_startfix$fix.arg, : Some
#> parameter names have no starting/fixed value but have a default value: mean.
#> The program is running, it'll take some time!
summary(res)
#> <div style='margin-bottom:0.5rem'><span style='color:#0b7285;font-weight:700'>Stratification summary</span></div>
#> <div><span style='color:#0b7285;font-weight:600'>Strata (h):</span> 2 &nbsp; <span style='color:#0b7285;font-weight:600'>n:</span> 150 &nbsp; <span style='color:#0b7285;font-weight:600'>N:</span> 487</div>
#> <div><span style='color:#0b7285;font-weight:600'>Data range:</span> [7.0000, 97.0000]  (d = 90.0000)</div>
#> <div><span style='color:#0b7285;font-weight:600'>Best-fit distribution:</span> weibull</div>
#> 
#> 
#> 
#> Table: Parameter estimate(s)
#> 
#> |Parameter |  Estimate|
#> |:---------|---------:|
#> |shape     |  3.371526|
#> |scale     | 59.909598|

Per-stratum results and totals

Strata	OSB	Wh	Vh	WhSh	nh	Nh	fh
1	53.44	0.49	114.72	5.278	74	240	0.31
2	97	0.51	116.01	5.463	76	247	0.31
Total		1.00	230.73	10.741	150	487	0.31

Similarly, in order to find the OSB and other quantities, we can apply the strata.distr() function to a Triangular population. Based on the assumption from past knowledge that the population follows Triangular distribution with the given attributes such as the initial value, distance, parameters, etc., we can execute the following command to obtain the results:

data(math)
final_marks <- math$final_marks
a <- min(final_marks); b <- max(final_marks)
a; b; d <- b - a; d
#> [1] 7
#> [1] 97
#> [1] 90

# Estimate parameters: fix min and max to sample bounds, estimate only mode by MLE
br <- pretty(final_marks, n = 20)
hh <- hist(final_marks, breaks = br, plot = FALSE)
m0 <- hh$mids[which.max(hh$counts)]          # histogram mode proxy
fit <- fitdist(final_marks, distr = "triang", method = "mle",
               start   = list(mode = m0),
               fix.arg = list(min = a, max = b))
#> Warning in checkparamlist(arg_startfix$start.arg, arg_startfix$fix.arg, : Some
#> parameter names have no starting/fixed value but have a default value: mean.
fit
#> Fitting of the distribution ' triang ' by maximum likelihood 
#> Parameters:
#>      estimate
#> mode     52.5
#> Fixed parameters:
#>     value
#> min     7
#> max    97

# 2-strata solution
res <- strata.distr(h=2, initval=7, dist=90, distr = "triangle",
      params = c(min=7, max=97, mode=54), n=150, N=352)
#> The program is running, it'll take some time!
summary(res)
#> <div style='margin-bottom:0.5rem'><span style='color:#0b7285;font-weight:700'>Stratification summary</span></div>
#> <div><span style='color:#0b7285;font-weight:600'>Strata (h):</span> 2 &nbsp; <span style='color:#0b7285;font-weight:600'>n:</span> 150 &nbsp; <span style='color:#0b7285;font-weight:600'>N:</span> 352</div>
#> <div><span style='color:#0b7285;font-weight:600'>Data range:</span> [7.0000, 97.0000]  (d = 90.0000)</div>
#> <div><span style='color:#0b7285;font-weight:600'>Best-fit distribution:</span> triangle</div>
#> 
#> 
#> 
#> Table: Parameter estimate(s)
#> 
#> |Parameter | Estimate|
#> |:---------|--------:|
#> |min       |        7|
#> |max       |       97|
#> |mode      |       54|

Per-stratum results and totals

Strata	OSB	Wh	Vh	WhSh	nh	Nh	fh
1	53	0.5	117.56	5.424	77	176	0.44
2	97	0.5	107.46	5.181	73	176	0.42
Total		1.0	225.01	10.604	150	352	0.43

8.3 Stratification for a Survey Variable with Right-Triangular Distribution

If a study variable follows the Right-Triangular distribution on the domain of [\(a, b\)], its two-parameter probability density function is given by:

\[\begin{equation} f(y) = \begin{cases} \dfrac{2(b-y)}{(b-a)^2}; & y \in [a, b]\\ 0; & otherwise \\ \end{cases} \tag{22} \end{equation}\]

where \(a\) is the location parameter and \(b\) is the scale parameter of the distribution.

Note that the Right-Triangular distribution is a special case of the Triangular distribution discussed in Section 7.2 where the parameters \(a=c\), i.e., minimium value is equal to the mode. Thus, the density function (22) is a two-parameter distribution.

8.3.1 DP Solution for Right-Triangular Distribution

To solve the MPP formulated for Right-Triangular distribution (22), we apply the algorithm using the DP technique discussed in Section 4. The recurrence relations used to determine the OSB are given by:

For the first stage, \(k=1\), at \(l_{1}^{*}=d_{1}\):

\[\begin{eqnarray} \Phi_{1}d_{1}&=& \textrm{Sqrt}\Biggl\{ \left[\dfrac{d_{1}(2(y_{0}-a)-d_{1})}{(b-a)^{2}}\right]^{2}\times\left[\dfrac{d_{1}^{2}(d_{1}^{2}-6(y_{0}-a)d_{1}+6a_{h}^{2})}{18(2(y_{0}-a)-d_{1})^{2}}\right]\Biggr\} \tag{23} \end{eqnarray}\]

And for the stages \(k\geq2\):

\[\begin{eqnarray} \Phi_{k}d_{k}&=&\min_{0\leq l_{k}\leq d_{k}}\Biggl\{ \textrm{Sqrt}\Biggl\{\left[\dfrac{l_{k}(2a_{k}-l_k)}{(b-a)^{2}}\right]^{2} \times\left[\dfrac{l_{k}^{2}(l_{k}^{2}-6a_{k}l_{k}+6a_{k}^{2})}{18(2a_{k}-l_{k})^{2}}\right]\Biggr\} + \Phi_{k-1}(d_{k}-l_{k})\Biggr\} \tag{24} \end{eqnarray}\]

where \(a_{k}=b-d_{k}+l_{k}-y_{0}\).

Substituting the values of \(a\), \(b\), \(y_{0}\) and \(d\), the OSW (\(l_{h}^{*}\)) and the OSB (\(y_{h}^{*}=y_{h-1}^{*}-l_{h}^{*}\)) are obtained by executing the strata.data() function.

8.3.2 A Numerical Example for Right-Triangular Distribution

A data following Right-Triangular distribution, of size \(N=5000\), was simulated to demonstrate the application of the stratifyR package. The simulated data takes in three parameters where \(a=c\) to indicate that it’s a Right-Triangular distribution. Upon fitting the data, its best-fit distribution exhibits a three-parameter Triangular distribution with the parameters \(min = 1.987776\), \(max=7.935599\) and \(mode=2.026685\). Note that because the \(min\) and \(mode\) parameters are very close to each and not exactly the same, it is treated as a Triangular distribution even thought the data simulated was for a Right-Triangular distribution. The minimum and maximum values in the simulated data are \([y_{0}, y_{L}] = [2.000052, 7.83871]\) with \(d=5.838658\).

To construct the OSB (\(h = 2\)) for the Right-Triangular data with a fixed total sample size of \(500\), we use the following code:

#Generate RT data
set.seed(12546)
data <- rtriangle(n=1000, a=2, b=8, c=2) #right-triangular since a=c
hist(data)

res <- strata.data(data, h = 2, n=500) # a 2-strata solution
#> Warning in checkparamlist(arg_startfix$start.arg, arg_startfix$fix.arg, : Some
#> parameter names have no starting/fixed value but have a default value: mean.
#> The program is running, it'll take some time!
summary(res)
#> <div style='margin-bottom:0.5rem'><span style='color:#0b7285;font-weight:700'>Stratification summary</span></div>
#> <div><span style='color:#0b7285;font-weight:600'>Strata (h):</span> 2 &nbsp; <span style='color:#0b7285;font-weight:600'>n:</span> 500 &nbsp; <span style='color:#0b7285;font-weight:600'>N:</span> 1000</div>
#> <div><span style='color:#0b7285;font-weight:600'>Data range:</span> [2.0001, 7.8387]  (d = 5.8387)</div>
#> <div><span style='color:#0b7285;font-weight:600'>Best-fit distribution:</span> lnorm</div>
#> 
#> 
#> 
#> Table: Parameter estimate(s)
#> 
#> |Parameter | Estimate|
#> |:---------|--------:|
#> |meanlog   | 1.327465|
#> |sdlog     | 0.353496|

Per-stratum results and totals

Strata	OSB	Wh	Vh	WhSh	nh	Nh	fh
1	4.17	0.58	0.38	0.359	246	582	0.42
2	7.84	0.42	0.78	0.369	254	418	0.61
Total		1.00	1.16	0.727	500	1000	0.50

We see in the above example that the simulated data is a Right-Triangular distribution, however, when data is fitted using MLE method in the package, it turns out that it best-fits Triangular distribution because the \(min\) is not exactly equal to \(mode\).

Similarly, in order to find the OSB and other quantities, we can apply the strata.distr() function to a Right-Triangular population. Based on the assumption from past knowledge that the population follows Right-Triangular distribution with the given attributes such as the initial value, distance, parameters, etc., we can execute the following command to obtain the results:

res <- strata.distr(h=2, initval=1.007202, dist=0.992781, distr = "rtriangle",
         params = c(min=2, max=10, mode=2), n=500, N=1000)
#> The program is running, it'll take some time!
summary(res)
#> <div style='margin-bottom:0.5rem'><span style='color:#0b7285;font-weight:700'>Stratification summary</span></div>
#> <div><span style='color:#0b7285;font-weight:600'>Strata (h):</span> 2 &nbsp; <span style='color:#0b7285;font-weight:600'>n:</span> 500 &nbsp; <span style='color:#0b7285;font-weight:600'>N:</span> 1000</div>
#> <div><span style='color:#0b7285;font-weight:600'>Data range:</span> [1.0072, 2.0000]  (d = 0.9928)</div>
#> <div><span style='color:#0b7285;font-weight:600'>Best-fit distribution:</span> rtriangle</div>
#> 
#> 
#> 
#> Table: Parameter estimate(s)
#> 
#> |Parameter | Estimate|
#> |:---------|--------:|
#> |min       |        2|
#> |max       |       10|
#> |mode      |        2|

Per-stratum results and totals

Strata	OSB	Wh	Vh	WhSh	nh	Nh	fh
1	1.5	-0.13	0.02	0.019	-129	-129	1.00
2	2	1.13	0.02	0.019	629	1129	0.56
Total		1.00	0.04	0.038	500	1000	0.50

The results show that this fits a two-paramter Right-Triangular distribution. Do note that in the above command, one has to specify all three parameters \(min=2, max=10\) and \(mode=2\), even for a Right-Triangular distribution, where \(min=mode\).

8.4 Stratification for a Survey Variable with Weibull Distribution

If the study variable follows the Weibull distribution on the interval \([y_{0}, y_{L}]\), its two-parameter probability density function with a state space \(y\geq 0\) is given by: \[\begin{equation} f(y; \theta, r)=\dfrac{r}{\theta}\left(\dfrac{y}{\theta}\right)^{r-1}e^{-\left(y/\theta\right)^{r}}, \;\;\;\;\;y\geq 0 \tag{25} \end{equation}\]

where \(r > 0\) is the shape parameter and \(\theta > 0\) is the scale parameter of the distribution.

8.4.1 DP Solution for Weibull Distribution

To solve the MPP formulated for Weibull distribution (25), we apply the algorithm using the DP technique discussed in Section 4. The recurrence relations involve the Upper Regularized Incomplete Gamma function \(Q(s, z)\), defined as: \[\begin{equation} Q(s,z)=\dfrac{1}{\Gamma(s)}\int_{z}^{\infty}t^{s-1}e^{-t}\,dt,\;\;\;\;\;s,z>0,\tag{25a} \end{equation}\] which in R is computed as . The recurrence relations used to determine the OSB are given by:

For the first stage, \(k=1\), at \(l_{1}^{*}=d_{1}\):

\[\begin{eqnarray} \Phi_{1}d_{1}&=& \textrm{Sqrt}\Biggl\{ \theta^{2}\,\Gamma\left(1+\dfrac{2}{r} \right)\left[e^{-\left(\frac{y_{0}}{\theta}\right)^{r}}-e^{-\left(\frac{d_{1}+y_{0}}{\theta}\right)^{r}}\right] \nonumber \\[10pt] &&\times \left[Q\left(1+\dfrac{2}{r},\left(\dfrac{y_{0}}{\theta}\right)^{r}\right)- Q\left(1+\dfrac{2}{r},\left(\dfrac{d_{1}+y_{0}}{\theta}\right)^{r}\right)\right]\nonumber\\[10pt] &&-\theta^{2}\left[\Gamma\left(1+\dfrac{1}{r}\right)\left [Q\left(1+\dfrac{1}{r},\left(\dfrac{y_{0}}{\theta}\right)^{r}\right)\right.\right.\nonumber\\[10pt] &&-Q\left.\left.\left(1+\dfrac{1}{r},\left(\dfrac{d_{1}+y_{0}}{\theta}\right)^{r}\right)\right]\right]^{2}\Biggr\} \tag{26} \end{eqnarray}\]

And for the stages \(k\geq2\):

\[\begin{eqnarray} \Phi_{k}d_{k}&=&\min_{0\leq l_{k}\leq d_{k}}\Biggl\{ \textrm{Sqrt}\Biggl\{ \theta^{2}\,\Gamma\left(1+\dfrac{2}{r} \right)\left[e^{-\left(\frac{d_{k}-l_{k}+y_{0}}{\theta}\right)^{r}}-e^{-\left(\frac{d_{k}+y_{0}}{\theta}\right)^{r}}\right]\nonumber\\[10pt] &&\times \left[Q\left(1+\dfrac{2}{r},\left(\dfrac{d_{k}-l_{k}+y_{0}}{\theta}\right)^{r}\right)- Q\left(1+\dfrac{2}{r},\left(\dfrac{d_{k}+y_{0}}{\theta}\right)^{r}\right)\right]\nonumber\\[15pt] &&-\theta^{2}\left[\Gamma\left(1+\dfrac{1}{r}\right)\left [Q\left(1+\dfrac{1}{r},\left(\dfrac{d_{k}-l_{k}+y_{0}}{\theta}\right)^{r}\right)\right.\right.\nonumber\\[15pt] &&-Q\left.\left.\left(1+\dfrac{1}{r},\left(\dfrac{d_{k}+y_{0}}{\theta}\right)^{r}\right)\right]\right]^{2}\Biggr\}+\Phi_{k-1}(d_{k}-l_{k})\Biggr\}. \tag{27} \end{eqnarray}\]

The recurrence relations (26) and (27) are solved using the DP technique to determine the OSB.

8.4.2 A Numerical Example for Weibull Distribution

A health data of size \(N=724\), called ‘anaemia’ data (Reddy et al. (2014)), is used to demonstrate the application of the stratifyR package on Weibull population. The ‘anaemia’ data comes from the National Nutritional Survey on the ``Micronutrient Status of Women in Fiji” and has many variables such as level of Iron, Folate, Zinc, etc. In this example, the variable Iron is used since it exhibits a 2-parameter Weibull distribution with the shape and scale parameters as \(r = 2.144586\) and \(\theta = 13.790744\) respectively. The minimum and maximum values are \([y_{0}, y_{L}] = [1.5, 34.7]\), which implies that \(d=33.2\).

To construct the OSB (\(h = 2\)) for the Iron data with a fixed total sample size of \(500\), we use the following codes:

data(anaemia) #using the anaemia data
Iron <- anaemia$Iron
hist(Iron)

min_value <- min(Iron) # Find the minimum value
Iron[Iron == min_value] <- -0.001 # Replace the minimum value with -0.001
res <- strata.data(Iron, h = 2, n=500) # a 2-strata solution
#> Warning in checkparamlist(arg_startfix$start.arg, arg_startfix$fix.arg, : Some
#> parameter names have no starting/fixed value but have a default value: mean.
#> The program is running, it'll take some time!
summary(res)
#> <div style='margin-bottom:0.5rem'><span style='color:#0b7285;font-weight:700'>Stratification summary</span></div>
#> <div><span style='color:#0b7285;font-weight:600'>Strata (h):</span> 2 &nbsp; <span style='color:#0b7285;font-weight:600'>n:</span> 500 &nbsp; <span style='color:#0b7285;font-weight:600'>N:</span> 724</div>
#> <div><span style='color:#0b7285;font-weight:600'>Data range:</span> [-0.0010, 34.7000]  (d = 34.7010)</div>
#> <div><span style='color:#0b7285;font-weight:600'>Best-fit distribution:</span> norm</div>
#> 
#> 
#> 
#> Table: Parameter estimate(s)
#> 
#> |Parameter |  Estimate|
#> |:---------|---------:|
#> |mean      | 12.194472|
#> |sd        |  6.026547|

Per-stratum results and totals

Strata	OSB	Wh	Vh	WhSh	nh	Nh	fh
1	12.58	0.58	8.83	1.724	244	420	0.58
2	34.7	0.42	18.49	1.805	256	304	0.84
Total		1.00	27.32	3.530	500	724	0.69

Similarly, in order to find the OSB and other quantities, we can apply the strata.distr() function to a Weibull population. Based on the assumption from past knowledge that the population follows Weibull distribution with the given attributes such as the initial value, distance, parameters, etc., we can execute the following command to obtain the results:

res <- strata.distr(h=2, initval=2.9, dist=55.9, distr = "weibull",
       params = c(shape=2.144586, scale=13.790744), n=500, N=5000)
#> The program is running, it'll take some time!
summary(res)
#> <div style='margin-bottom:0.5rem'><span style='color:#0b7285;font-weight:700'>Stratification summary</span></div>
#> <div><span style='color:#0b7285;font-weight:600'>Strata (h):</span> 2 &nbsp; <span style='color:#0b7285;font-weight:600'>n:</span> 500 &nbsp; <span style='color:#0b7285;font-weight:600'>N:</span> 5000</div>
#> <div><span style='color:#0b7285;font-weight:600'>Data range:</span> [2.9000, 58.8000]  (d = 55.9000)</div>
#> <div><span style='color:#0b7285;font-weight:600'>Best-fit distribution:</span> weibull</div>
#> 
#> 
#> 
#> Table: Parameter estimate(s)
#> 
#> |Parameter |  Estimate|
#> |:---------|---------:|
#> |shape     |  2.144586|
#> |scale     | 13.790744|

Per-stratum results and totals

Strata	OSB	Wh	Vh	WhSh	nh	Nh	fh
1	13.05	0.59	7.49	1.516	239	2943	0.08
2	58.8	0.41	16.28	1.660	261	2057	0.13
Total		1.00	23.77	3.176	500	5000	0.10

8.5 Stratification for a Survey Variable with Gamma Distribution

If the study variable follows the Gamma distribution (i.e., \(y\sim \Gamma(r, \theta)\)) on the interval \([y_{0}, y_{L}]\), it has the following two-parameter probability density function:

\[\begin{equation} f(y;r,\theta)=\dfrac{1}{\theta^{r}\Gamma(r)}\,y^{r-1}e^{-\dfrac{y}{\theta}},\;\;\;\;\;\;\;y>0;\;\;r,\theta>0, \tag{28} \end{equation}\]

where is a shape parameter and \(\theta\) is the scale parameter and \(\Gamma(r)\) is a Gamma function defined by

\[\begin{equation}\label{222} \Gamma(r)=\int_{0}^{\infty}t^{r-1}e^{-t}\,dt,\;\;\;\;\;\; r > 0. \tag{29} \end{equation}\]

The function in equation \((\ref{222})\) is also defined by an upper incomplete gamma function \(\Gamma(r,x)\) and a lower incomplete gamma function \(\gamma(r,x)\), respectively, as follows:

\[\begin{eqnarray} \Gamma(r,y)=\int_{y}^{\infty}t^{r-1}e^{-t}\,dt;\label{223} \tag{30}\\ \gamma(r,y)=\int_{0}^{y}t^{r-1}e^{-t}\,dt.\label{224} \tag{31} \end{eqnarray}\]

There also exist regularized/normalised incomplete Gamma functions which give a value restricted between \(0\) and \(1\) and can be stated as:

\[\begin{eqnarray} Q(r,y)=\dfrac{1}{\Gamma(r)}\int_{y}^{\infty}t^{r-1}e^{-t}\,dt,\;\;\;\;\;\;r,y>0;\;\;\Gamma(r)\neq0;\label{225} \tag{32}\\ P(r,y)=\dfrac{1}{\Gamma(r)}\int_{0}^{y}t^{r-1}e^{-t}\,dt,\;\;\;\;\;\;r,y>0;\;\;\Gamma(r)\neq0,\label{226} \tag{33} \end{eqnarray}\]

where \(Q(r,y)\) denotes the Upper Regularized Incomplete Gamma function while \(P(r,y)\) denotes the Lower Regularized Incomplete Gamma function (Abramowitz and Stegun (1972), Chapter 6). Note that \(Q(r,y)=1-P(r,y)\). In R, \(Q(r,y)\) is evaluated as , which is the regularized form. It is important to note that \(Q(r,y)\) is distinct from the unregularized upper incomplete gamma function \(\Gamma(r,y)\) (equation 30); they are related by \(Q(r,y)=\Gamma(r,y)/\Gamma(r)\). The DP recurrence relations for both Weibull (equations 26–27) and Gamma (equations 34–35) use the regularized \(Q(r,y)\), and stratifyR 2.0-1 implements these correctly using .

8.5.1 DP Solution for Gamma Distribution

To solve the MPP formulated for Gamma distribution (28), we apply the algorithm using the DP technique discussed in Section 4. The recurrence relations used to determine the OSB are given by:

For the first stage, \(k=1\), at \(l_{1}^{*}=d_{1}\):

\[\begin{eqnarray} \Phi_{1}d_{1}&=& \textrm{Sqrt}\Biggl\{ \theta^{2}\,r(r+1)\left[ Q\left(r,\dfrac{y_{0}}{\theta}\right)-Q\left(r,\dfrac{d_{1}+y_{0}}{\theta}\right)\right]\nonumber\\[10pt] &&\times\left[Q\left(r+2,\dfrac{y_{0}}{\theta}\right)-Q\left(r+2,\dfrac{d_{1}+y_{0}}{\theta}\right) \right]\nonumber\\[10pt] &&-\theta^{2}r^{2}\left[ Q\left(r+1,\dfrac{y_{0}}{\theta}\right)-Q\left(r+1,\dfrac{d_{1}+y_{0}}{\theta}\right)\right]^{2}\Biggr\} \tag{34} \end{eqnarray}\]

And for the stages \(k\geq2\):

\[\begin{eqnarray} \Phi_{k}d_{k}&=&\min_{0\leq l_{k}\leq d_{k}}\Biggl\{ \textrm{Sqrt}\Biggl\{ \theta^{2}\,r(r+1)\left[ Q\left(r,\dfrac{d_{k}-l_{k}+y_{0}}{\theta}\right.\right)\nonumber\\[10pt] &&-\left.Q\left(r,\dfrac{d_{k}+y_{0}}{\theta}\right)\right] \times\left[ Q\left(r+2,\dfrac{d_{k}-l_{k}+y_{0}}{\theta}\right.\right)\nonumber\\[10pt] &&-\left.Q\left(r+2,\dfrac{d_{k}+y_{0}}{\theta}\right) \right]-\theta^{2}r^{2}\times\left[ Q\left(r+1,\dfrac{d_{k}-l_{k}+y_{0}}{\theta}\right)\right.\nonumber\\[10pt] &&-\left.Q\left(r+1,\dfrac{d_{k}+y_{0}}{\theta}\right)\right]^{2}\Biggr\}+\Phi_{k-1}(d_{k}-l_{k})\Biggr\}. \tag{35} \end{eqnarray}\]

The recurrence relations (34) and (35) are solved using the DP technique to determine the OSB.

8.5.2 A Numerical Example for Gamma Distribution

Again, the health data of size \(N=724\), derived from the ‘National Nutritional Survey’ on the ‘Micronutrient Status of Women in Fiji’ is used to demonstrate the application of the stratifyR package on Gamma population. In this example, the variable Folate is used since it exhibits a 2-parameter Gamma distribution with the shape and scale parameters as \(r = 6.9922\) and \(\theta = 2.5785\) respectively. The minimum and maximum values are \([y_{0}, y_{L}] = [4.9, 45.4]\), which implies that \(d=40.5\).

To construct the OSB (\(h = 2\)) for the Folate data with a fixed total sample size of \(500\), we use the following codes:

data(anaemia)
Folate <- anaemia$Folate
hist(Folate)

min_value <- min(Folate) # Find the minimum value
Folate[Folate == min_value] <- -0.001 # Replace the minimum value with -0.001
res <- strata.data(Folate, h = 2, n=500) # a 2-strata solution
#> Warning in checkparamlist(arg_startfix$start.arg, arg_startfix$fix.arg, : Some
#> parameter names have no starting/fixed value but have a default value: mean.
#> The program is running, it'll take some time!
summary(res)
#> <div style='margin-bottom:0.5rem'><span style='color:#0b7285;font-weight:700'>Stratification summary</span></div>
#> <div><span style='color:#0b7285;font-weight:600'>Strata (h):</span> 2 &nbsp; <span style='color:#0b7285;font-weight:600'>n:</span> 500 &nbsp; <span style='color:#0b7285;font-weight:600'>N:</span> 724</div>
#> <div><span style='color:#0b7285;font-weight:600'>Data range:</span> [-0.0010, 45.4000]  (d = 45.4010)</div>
#> <div><span style='color:#0b7285;font-weight:600'>Best-fit distribution:</span> norm</div>
#> 
#> 
#> 
#> Table: Parameter estimate(s)
#> 
#> |Parameter |  Estimate|
#> |:---------|---------:|
#> |mean      | 18.017124|
#> |sd        |  6.985322|

Per-stratum results and totals

Strata	OSB	Wh	Vh	WhSh	nh	Nh	fh
1	18.13	0.56	9.85	1.752	210	404	0.52
2	45.4	0.44	29.95	2.419	290	320	0.91
Total		1.00	39.80	4.170	500	724	0.69

Similarly, in order to find the OSB and other quantities, we can apply the strata.distr() function to a Gamma population. Based on the assumption from past knowledge that the population follows Gamma distribution with the given attributes such as the initial value, distance, parameters, etc., we can execute the following command to obtain the results:

res <- strata.distr(h=2, initval=0.5, dist=50, distr = "gamma",
       params = c(shape=3.835768, rate=0.340328), n=500, N=12000)
#> The program is running, it'll take some time!
summary(res)
#> <div style='margin-bottom:0.5rem'><span style='color:#0b7285;font-weight:700'>Stratification summary</span></div>
#> <div><span style='color:#0b7285;font-weight:600'>Strata (h):</span> 2 &nbsp; <span style='color:#0b7285;font-weight:600'>n:</span> 500 &nbsp; <span style='color:#0b7285;font-weight:600'>N:</span> 12000</div>
#> <div><span style='color:#0b7285;font-weight:600'>Data range:</span> [0.5000, 50.5000]  (d = 50.0000)</div>
#> <div><span style='color:#0b7285;font-weight:600'>Best-fit distribution:</span> gamma</div>
#> 
#> 
#> 
#> Table: Parameter estimate(s)
#> 
#> |Parameter | Estimate|
#> |:---------|--------:|
#> |shape     | 3.835768|
#> |rate      | 0.340328|

Per-stratum results and totals

Strata	OSB	Wh	Vh	WhSh	nh	Nh	fh
1	12.18	0.63	6.95	1.653	247	7524	0.03
2	50.5	0.37	20.49	1.688	253	4476	0.06
Total		1.00	27.45	3.342	500	12000	0.04

8.6 Stratification for a Survey Variable with Exponential Distribution

If the study variable follows the Exponential distribution on the interval \([y_{0}, y_{L}]\), its one-parameter probability density function is given by:

\[\begin{equation} f(y; \lambda)= \begin{cases} \lambda\mathrm{e}^{-\lambda y}; & y > 0\\ \;\;\;\;0; & elsewhere \\ \end{cases} \tag{36} \end{equation}\]

where \(\lambda\) is the continuous rate parameter (or the inverse scale parameter).

8.6.1 DP Solution for Exponential Distribution

To solve the MPP formulated for Exponential distribution (36), we apply the algorithm using the DP technique discussed in Section 4. The recurrence relations used to determine the OSB are given by:

For the first stage, \(k=1\), at \(l_{1}^{*}=d_{1}\):

\[\begin{eqnarray} \Phi_{1}d_{1}&=& \textrm{Sqrt}\Biggl\{ \left(e^{-\lambda y_{0}}\right)^{2}\left[\dfrac{1}{\lambda^2}\left(1-e^{-\lambda l_{k}}\right)^{2}-l_{k}^{2}e^{-\lambda l_{k}}\right]\Biggr\} \tag{37} \end{eqnarray}\]

And for the stages \(k\geq2\):

\[\begin{eqnarray} \Phi_{k}d_{k}&=&\min_{0\leq l_{k}\leq d_{k}}\Biggl\{ \textrm{Sqrt}\Biggl\{ \left(e^{-\lambda (d_{k}-l_{k}+y_{0})}\right)^{2}\left[\dfrac{1}{\lambda^2}\left(1-e^{-\lambda l_{k}}\right)^{2}-l_{k}^{2}e^{-\lambda l_{k}}\right]\Biggr\} + \Phi_{k-1}(d_{k}-l_{k})\Biggr\} \tag{38} \end{eqnarray}\]

Substituting the values of \(\lambda\), \(y_{0}\) and \(d\), the OSW (\(l_{h}^{*}\)) and the OSB (\(y_{h}^{*}=y_{h-1}^{*}-l_{h}^{*}\)) are obtained by executing the strata.data() function.

8.6.2 A Numerical Example for Exponential Distribution

A data following Exponential distribution (herein called \(Exp\) data), of size \(N=10000\) was simulated to demonstrate the application of the stratifyR package on an Exponential population. The data exhibits an Exponential distribution with the parameter \(rate = 1.359205\). The minimum and maximum values in the simulated data are \([y_{0}, y_{L}] = [5.747904e-05, 8.016871]\), which implies that \(d=8.016814\).

To construct the OSB (\(h = 2\)) for the data that follows Exponential distribution with a fixed total sample size of \(500\), we use the following codes:

set.seed(28951)
data <- rexp(5000, rate = 1.36)
hist(data)

res <- strata.data(data, h = 2, n=500) # a 2-strata solution
#> Warning in checkparamlist(arg_startfix$start.arg, arg_startfix$fix.arg, : Some
#> parameter names have no starting/fixed value but have a default value: mean.
#> The program is running, it'll take some time!
#> Warning in mledist(data, distname, start = arg_startfix$start.arg, fix.arg =
#> arg_startfix$fix.arg, : The BFGS method cannot be used with bounds without
#> provided the gradient. The method is changed to L-BFGS-B.
summary(res)
#> <div style='margin-bottom:0.5rem'><span style='color:#0b7285;font-weight:700'>Stratification summary</span></div>
#> <div><span style='color:#0b7285;font-weight:600'>Strata (h):</span> 2 &nbsp; <span style='color:#0b7285;font-weight:600'>n:</span> 500 &nbsp; <span style='color:#0b7285;font-weight:600'>N:</span> 5000</div>
#> <div><span style='color:#0b7285;font-weight:600'>Data range:</span> [0.0001, 8.0169]  (d = 8.0168)</div>
#> <div><span style='color:#0b7285;font-weight:600'>Best-fit distribution:</span> exp</div>
#> 
#> 
#> 
#> Table: Parameter estimate(s)
#> 
#> |Parameter | Estimate|
#> |:---------|--------:|
#> |rate      | 1.359205|

Per-stratum results and totals

Strata	OSB	Wh	Vh	WhSh	nh	Nh	fh
1	0.93	0.72	0.07	0.188	235	3600	0.07
2	8.02	0.28	0.57	0.211	265	1400	0.19
Total		1.00	0.64	0.399	500	5000	0.10

Similarly, in order to find the OSB and other quantities, we can apply the strata.distr() function to a Exponential population. Based on the assumption from past knowledge that the population follows Exponential distribution with the given attributes such as the initial value, distance, parameters, etc., we can execute the following command to obtain the results:

set.seed(28951)
data <- rexp(5000, rate = 1.36)
min(data); max(data); d=max(data)-min(data);d
#> [1] 5.747904e-05
#> [1] 8.016871
#> [1] 8.016814
fit <- fitdist(data, distr="exp", method="mle")
fit
#> Fitting of the distribution ' exp ' by maximum likelihood 
#> Parameters:
#>      estimate Std. Error
#> rate 1.359205 0.01922205
res <- strata.distr(h=2, initval=5.748e-05, dist=8.017, distr = "exp", 
             params = c(rate=1.36), n=500, N=5000) #a 2-strata solution
#> The program is running, it'll take some time!
summary(res)
#> <div style='margin-bottom:0.5rem'><span style='color:#0b7285;font-weight:700'>Stratification summary</span></div>
#> <div><span style='color:#0b7285;font-weight:600'>Strata (h):</span> 2 &nbsp; <span style='color:#0b7285;font-weight:600'>n:</span> 500 &nbsp; <span style='color:#0b7285;font-weight:600'>N:</span> 5000</div>
#> <div><span style='color:#0b7285;font-weight:600'>Data range:</span> [0.0001, 8.0171]  (d = 8.0170)</div>
#> <div><span style='color:#0b7285;font-weight:600'>Best-fit distribution:</span> exp</div>
#> 
#> 
#> 
#> Table: Parameter estimate(s)
#> 
#> |Parameter | Estimate|
#> |:---------|--------:|
#> |rate      |     1.36|

Per-stratum results and totals

Strata	OSB	Wh	Vh	WhSh	nh	Nh	fh
1	0.93	0.72	0.07	0.185	236	3589	0.07
2	8.02	0.28	0.54	0.207	264	1411	0.19
Total		1.00	0.60	0.392	500	5000	0.10

8.7 Stratification for a Survey Variable with Uniform Distribution

If the study variable follows the Uniform distribution on the interval \([y_{0}, y_{L}]\), its two-parameter probability density function is given by:

\[\begin{equation} f(y; a,b)= \begin{cases} \dfrac{1}{b-a}; & y > 0\\ \;\;\;\;0; & otherwise \\ \end{cases} \tag{39} \end{equation}\]

where \(a\) and \(b\) are the continuous boundary parameters, i.e., \(a\) and \(b\) are the minimum and maximum values respectively.

8.7.1 DP Solution for Uniform Distribution

To solve the MPP formulated for Uniform distribution (39), we apply the algorithm using the DP technique discussed in Section 4. The recurrence relations used to determine the OSB are given by:

For the first stage, \(k=1\), at \(l_{1}^{*}=d_{1}\):

\[\begin{eqnarray} \Phi_{1}d_{1}&=& \textrm{Sqrt}\Biggl\{ \sum\limits_{h=1}^{L} \dfrac{d_{1}^{2}}{2\sqrt{3}(b-a)}\Biggr\} \tag{40} \end{eqnarray}\]

And for the stages \(k\geq2\):

\[\begin{eqnarray} \Phi_{k}d_{k}&=&\min_{0\leq l_{k}\leq d_{k}}\Biggl\{ \textrm{Sqrt}\Biggl\{ \sum\limits_{h=1}^{L} \dfrac{l_{k}^{2}}{2\sqrt{3}(b-a)}\Biggr\}+ \Phi_{k-1}(d_{k}-l_{k})\Biggr\} \tag{41} \end{eqnarray}\]

Substituting the values of \(a\), \(b\), \(y_{0}\) and \(d\), the OSW (\(l_{h}^{*}\)) and the OSB (\(y_{h}^{*}=y_{h-1}^{*}-l_{h}^{*}\)) are obtained by executing the strata.data() function.

8.7.2 A Numerical Example for Uniform Distribution

A data following Uniform distribution of size \(N=5000\) was simulated to demonstrate the application of the stratifyR package on a Uniform population. The data for Uniform distribution is simulated with the parameters \(min = 2\) and \(max=15\). When fitted, the minimum and maximum values in the simulated data are \([y_{0}, y_{L}] = [2.006522, 14.99764]\), which implies that \(d=12.99112\).

To construct the OSB (\(h = 2\)) for the Uniformly-distributed data with a fixed total sample size of \(450\), we use the following codes:

set.seed(15669)
data <- runif(5000, min = 2, max = 15)
hist(data)

res <- strata.data(data, h = 2, n=450) # a 2-strata solution
#> Warning in checkparamlist(arg_startfix$start.arg, arg_startfix$fix.arg, : Some
#> parameter names have no starting/fixed value but have a default value: mean.
#> The program is running, it'll take some time!
summary(res)
#> <div style='margin-bottom:0.5rem'><span style='color:#0b7285;font-weight:700'>Stratification summary</span></div>
#> <div><span style='color:#0b7285;font-weight:600'>Strata (h):</span> 2 &nbsp; <span style='color:#0b7285;font-weight:600'>n:</span> 450 &nbsp; <span style='color:#0b7285;font-weight:600'>N:</span> 5000</div>
#> <div><span style='color:#0b7285;font-weight:600'>Data range:</span> [2.0065, 14.9976]  (d = 12.9911)</div>
#> <div><span style='color:#0b7285;font-weight:600'>Best-fit distribution:</span> unif</div>
#> 
#> 
#> 
#> Table: Parameter estimate(s)
#> 
#> |Parameter |  Estimate|
#> |:---------|---------:|
#> |min       |  2.006522|
#> |max       | 14.997645|

Per-stratum results and totals

Strata	OSB	Wh	Vh	WhSh	nh	Nh	fh
1	8.5	0.49	3.55	0.919	220	2437	0.09
2	15	0.51	3.52	0.962	230	2563	0.09
Total		1.00	7.07	1.880	450	5000	0.09

Note that the above results indicate that the best-fit distribution is Triangular and not Uniform. This is because when stratifyR package assesses the data to ascertain the best-fit distribution, it is not able to calculate its AIC value for Uniform distribution. Hence, the distribution that gives the lowest AIC is taken to be the best-fit distribution. Since AIC for Uniform distribution is not available, the next best-fit distribution (Triangular) is be chosen. The results, you would find, are still quite accurate!

Similarly, we can apply the strata.distr() function for the Uniform distribution where the arguments can be assumed from past knowledge. To construct the OSB for \(h = 2\) for a hypothetical variable that follows Uniform distribution (with parameters \(min=3\) and \(max=15\)) with a fixed total sample size of \(450\) from a population of \(5000\), the following command is used:

# For a hypothetical uniform distribution, it does give a result
res <- strata.distr(h=2, initval=3, dist=12, distr = "unif",
                 params = c(min=3, max=15), n=450, N=5000)
#> The program is running, it'll take some time!
summary(res)
#> <div style='margin-bottom:0.5rem'><span style='color:#0b7285;font-weight:700'>Stratification summary</span></div>
#> <div><span style='color:#0b7285;font-weight:600'>Strata (h):</span> 2 &nbsp; <span style='color:#0b7285;font-weight:600'>n:</span> 450 &nbsp; <span style='color:#0b7285;font-weight:600'>N:</span> 5000</div>
#> <div><span style='color:#0b7285;font-weight:600'>Data range:</span> [3.0000, 15.0000]  (d = 12.0000)</div>
#> <div><span style='color:#0b7285;font-weight:600'>Best-fit distribution:</span> unif</div>
#> 
#> 
#> 
#> Table: Parameter estimate(s)
#> 
#> |Parameter | Estimate|
#> |:---------|--------:|
#> |min       |        3|
#> |max       |       15|

Per-stratum results and totals

Strata	OSB	Wh	Vh	WhSh	nh	Nh	fh
1	9	0.5	3	0.866	225	2500	0.09
2	15	0.5	3	0.866	225	2500	0.09
Total		1.0	6	1.732	450	5000	0.09

8.8 Stratification for a Survey Variable with Normal Distribution

If the study variable follows the Normal distribution on the interval \([y_{0}, y_{L}]\), it has the following two-parameter probability density function: \[\begin{equation} f(y;\mu, \sigma)=\dfrac{1}{\sigma\sqrt{2\pi}}\,exp\left\{-\dfrac{1}{2}\left(\dfrac{y-\mu}{\sigma}\right)^{2}\right\},\;\;\;\;\;\ -\infty < y < \infty \tag{42} \end{equation}\]

where \(\sigma > 0\) is a scale parameter and \(\mu\) is the location parameter.

The following definitions of error function are worth noting since they are needed to simplify the integrations used to derive the stratum weight, mean and variance due to normal distribution. \[\begin{eqnarray} erf(z) = \dfrac{2}{\sqrt{\pi}}\int_{0}^{z}exp\left\{-y^{2}\right\}\,dy \tag{43} \end{eqnarray}\] It can also be written as \[\begin{eqnarray}\label{erf} \dfrac{1}{\sqrt{2\pi}}\int_{0}^{z}exp\left\{-\dfrac{1}{2}y^{2}\right\}\,dy = \dfrac{1}{2}\,erf\left(\dfrac{z}{\sqrt{2}}\right) \tag{44} \end{eqnarray}\]

8.8.1 DP Solution for Normal Distribution

To solve the MPP formulated for Normal distribution (42), we apply the algorithm using the DP technique discussed in Section 4. The recurrence relations used to determine the OSB are given by:

For the first stage, \(k=1\), at \(l_{1}^{*}=d_{1}\):

\[\begin{eqnarray} \Phi_{1}d_{1}&=& \textrm{Sqrt}\Biggl\{ \dfrac{\sigma^{2}}{2\sqrt{2\pi}} \left[erf\left(\dfrac{d_{1}+y_{0}-\mu}{\sigma\sqrt{2}}\right) - erf\left(\dfrac{y_{0}-\mu}{\sigma\sqrt{2}}\right)\right]\nonumber \\[10pt] && \times\left[\left(\dfrac{y_{0}-\mu}{\sigma}\right)exp\left(-\left(\dfrac{y_{0}-\mu}{\sigma\sqrt{2}}\right)^{2}\right)- \left(\dfrac{d_{1}+y_{0}-\mu}{\sigma}\right)exp\left(-\left(\dfrac{d_{1}+y_{0}-\mu}{\sigma\sqrt{2}}\right)^{2}\right)\right]\nonumber\\[10pt] && + \dfrac{\sigma^{2}}{4}\left[erf\left(\dfrac{d_{1}+y_{0}-\mu}{\sigma\sqrt{2}}\right) - erf\left(\dfrac{y_{0}-\mu}{\sigma\sqrt{2}}\right)\right]^{2}\nonumber\\[10pt] && - \dfrac{\sigma^{2}}{2\pi}\left[exp\left(-\left(\dfrac{y_{0}-\mu}{\sigma\sqrt{2}}\right)^{2}\right)- exp\left(-\left(\dfrac{d_{1}+y_{0}-\mu}{\sigma\sqrt{2}}\right)^{2}\right)\right]^{2}\Biggr\} \tag{45} \end{eqnarray}\]

And for the stages \(k\geq2\):

\[\begin{eqnarray} \Phi_{k}d_{k}&=&\min_{0\leq l_{k}\leq d_{k}}\Biggl\{ \textrm{Sqrt}\Biggl\{ \dfrac{\sigma^{2}}{2\sqrt{2\pi}} \left[ erf\left(\dfrac{d_{k}+y_{0}-\mu}{\sigma\sqrt{2}}\right) - erf\left(\dfrac{(d_{k}-l_{k}+y_{0})-\mu}{\sigma\sqrt{2}}\right) \right]\nonumber \\[10pt] && \times\left[\left(\dfrac{(d_{k}-l_{k}+y_{0})-\mu}{\sigma}\right)exp\left(-\left(\dfrac{(d_{k}-l_{k}+y_{0})-\mu}{\sigma\sqrt{2}}\right)^{2}\right)\right. \nonumber\\[10pt] && - \left. \left(\dfrac{d_{k}+y_{0}-\mu}{\sigma}\right)exp\left(-\left(\dfrac{d_{k}+y_{0}-\mu}{\sigma\sqrt{2}}\right)^{2}\right)\right]\nonumber\\[10pt] && + \dfrac{\sigma^{2}}{4}\left[erf\left(\dfrac{d_{k}+y_{0}-\mu}{\sigma\sqrt{2}}\right) - erf\left(\dfrac{(d_{k}-l_{k}+y_{0})-\mu}{\sigma\sqrt{2}}\right)\right]^{2}\nonumber\\[10pt] && - \dfrac{\sigma^{2}}{2\pi}\left[exp\left(-\left(\dfrac{(d_{k}-l_{k}+y_{0})-\mu}{\sigma\sqrt{2}}\right)^{2}\right)-exp\left(-\left(\dfrac{d_{k}+y_{0}-\mu}{\sigma\sqrt{2}}\right)^{2}\right)\right]^{2}\Biggr\} + \Phi_{k-1}(d_{k}-l_{k})\Biggr\} \tag{46} \end{eqnarray}\]

Substituting the values of \(\mu\), \(\sigma\), \(y_{0}\) and \(d\), the OSW (\(l_{h}^{*}\)) and the OSB (\(y_{h}^{*}=y_{h-1}^{*}-l_{h}^{*}\)) are obtained by executing the strata.data() function.

8.8.2 A Numerical Example for Normal Distribution

A data following Normal distribution (herein called \(Norm\) data), of size \(N=5000\) was simulated to demonstrate the application of the stratifyR package on a Normal population. The data exhibits an Normal distribution with the parameters \(mean = 16.010776\) and \(sd=1.662357\). The minimum and maximum values in the simulated data are \([y_{0}, y_{L}] = [9.923816, 22.51267]\), which implies that \(d=10.62118\).

To construct the OSB for \(h = 2\) using the \(Norm\) data with a fixed total sample size of \(580\), the command below can be used:

set.seed(89821)
data <- rnorm(5000, mean = 16, sd = 1.65)
hist(data)

res <- strata.data(data, h = 2, n=500) #construct a 2-strata solution
#> Warning in checkparamlist(arg_startfix$start.arg, arg_startfix$fix.arg, : Some
#> parameter names have no starting/fixed value but have a default value: mean.
#> The program is running, it'll take some time!
summary(res)
#> <div style='margin-bottom:0.5rem'><span style='color:#0b7285;font-weight:700'>Stratification summary</span></div>
#> <div><span style='color:#0b7285;font-weight:600'>Strata (h):</span> 2 &nbsp; <span style='color:#0b7285;font-weight:600'>n:</span> 500 &nbsp; <span style='color:#0b7285;font-weight:600'>N:</span> 5000</div>
#> <div><span style='color:#0b7285;font-weight:600'>Data range:</span> [9.9238, 22.5127]  (d = 12.5889)</div>
#> <div><span style='color:#0b7285;font-weight:600'>Best-fit distribution:</span> norm</div>
#> 
#> 
#> 
#> Table: Parameter estimate(s)
#> 
#> |Parameter |  Estimate|
#> |:---------|---------:|
#> |mean      | 16.010776|
#> |sd        |  1.662357|

Per-stratum results and totals

Strata	OSB	Wh	Vh	WhSh	nh	Nh	fh
1	16.01	0.5	0.98	0.493	245	2487	0.1
2	22.51	0.5	1.04	0.513	255	2512	0.1
Total		1.0	2.02	1.006	500	4999	0.1

Similarly, in order to find the OSB and other quantities, we can apply the strata.distr() function to a Normal population. Based on the assumption from past knowledge that the population follows Normal distribution with the given attributes such as the initial value, distance, parameters, etc., we can execute the following command to obtain the results:

set.seed(89821)
data <- rnorm(5000, mean = 16, sd = 1.65)
min(data); max(data); d=max(data)-min(data);d
#> [1] 9.923816
#> [1] 22.51267
#> [1] 12.58885
fit <- fitdist(data, distr="norm", method="mle")
fit
#> Fitting of the distribution ' norm ' by maximum likelihood 
#> Parameters:
#>       estimate Std. Error
#> mean 16.010776 0.02350928
#> sd    1.662357 0.01662355
res <- strata.distr(h=2, initval=9.923816, dist=12.58885, distr = "norm",
             params = c(mean=16.010776, sd=1.662357), n=500, N=5000)
#> The program is running, it'll take some time!
summary(res)
#> <div style='margin-bottom:0.5rem'><span style='color:#0b7285;font-weight:700'>Stratification summary</span></div>
#> <div><span style='color:#0b7285;font-weight:600'>Strata (h):</span> 2 &nbsp; <span style='color:#0b7285;font-weight:600'>n:</span> 500 &nbsp; <span style='color:#0b7285;font-weight:600'>N:</span> 5000</div>
#> <div><span style='color:#0b7285;font-weight:600'>Data range:</span> [9.9238, 22.5127]  (d = 12.5889)</div>
#> <div><span style='color:#0b7285;font-weight:600'>Best-fit distribution:</span> norm</div>
#> 
#> 
#> 
#> Table: Parameter estimate(s)
#> 
#> |Parameter |  Estimate|
#> |:---------|---------:|
#> |mean      | 16.010776|
#> |sd        |  1.662357|

Per-stratum results and totals

Strata	OSB	Wh	Vh	WhSh	nh	Nh	fh
1	16.01	0.5	1	0.499	250	2499	0.1
2	22.51	0.5	1	0.501	250	2501	0.1
Total		1.0	2	1.000	500	5000	0.1

8.9 Stratification for a Survey Variable with Log-Normal Distribution

If the study variable follows the Log-normal distribution on the interval \([y_{0}, y_{L}]\), it has the following two-parameter probability density function:

\[\begin{equation} f(y;\mu,\sigma)=\dfrac{1}{y\sigma\sqrt{2\pi}}\,exp\left\{-\dfrac{1}{2}\left(\dfrac{ln(y)-\mu}{\sigma}\right)^{2}\right\},\;\;\;\;\;\ y>0 \tag{47} \end{equation}\]

where \(\sigma > 0\) is a scale parameter and \(\mu\) is the location parameter.

8.9.1 DP Solution for Log-Normal Distribution

To solve the MPP formulated for Log-Normal distribution (47), we apply the algorithm using the DP technique discussed in Section 4. The recurrence relations used to determine the OSB are given by:

For the first stage, \(k=1\), at \(l_{1}^{*}=d_{1}\):

\[\begin{eqnarray} \Phi_{1}d_{1}&=& \textrm{Sqrt}\Biggl\{\frac{1}{4}exp\left(2\mu+2\sigma^{2}\right)\left[ erf\left(\frac{ln(d_{1}+y_{0})-\mu-2\sigma^{2}}{\sigma\sqrt{2}}\right)\right.\nonumber \\[10pt] && \left. - erf\left(\frac{ln(y_{0})-\mu-2\sigma^{2}}{\sigma\sqrt{2}}\right) \right] \left[ erf\left(\frac{ln(d_{1}+y_{0})-\mu}{\sigma\sqrt{2}}\right) - erf\left(\frac{ln(y_{0})-\mu}{\sigma\sqrt{2}}\right)\right]\nonumber \\[10pt] && -\frac{1}{4}exp\left(2\mu+\sigma^{2}\right)\left[erf\left(\frac{ln(d_{1}+y_{0})-\mu-\sigma^{2}}{\sigma\sqrt{2}}\right) - erf\left(\frac{ln(y_{0})-\mu-\sigma^{2}}{\sigma\sqrt{2}}\right) \right]^{2}\Biggr\} \tag{48} \end{eqnarray}\]

And for the stages \(k\geq2\):

\[\begin{eqnarray} \Phi_{k}d_{k}&=&\min_{0\leq l_{k}\leq d_{k}}\Biggl\{ \textrm{Sqrt}\Biggl\{ \frac{1}{4}exp\left(2\mu+2\sigma^{2}\right)\left[erf\left(\frac{ln(d_{k}+y_{0})-\mu-2\sigma^{2}}{\sigma\sqrt{2}}\right)\right.\nonumber \\[10pt] && - \left. erf\left(\frac{ln(d_{k}-l_{k}+y_{0})-\mu-2\sigma^{2}}{\sigma\sqrt{2}}\right) \right] \left[ erf\left(\frac{ln(d_{k}+y_{0})-\mu}{\sigma\sqrt{2}}\right)\right.\nonumber \\[10pt] && -\left. erf\left(\frac{ln(d_{k}-l_{k}+y_{0})-\mu}{\sigma\sqrt{2}}\right) \right]-\frac{1}{4}exp\left(2\mu+\sigma^{2}\right)\nonumber \\[10pt] &&\times \left[ erf\left(\frac{ln(d_{k}+y_{0})-\mu-\sigma^{2}}{\sigma\sqrt{2}}\right) - erf\left(\frac{ln(d_{k}-l_{k}+y_{0})-\mu-\sigma^{2}}{\sigma\sqrt{2}}\right) \right]^{2}\Biggr\} + \Phi_{k-1}(d_{k}-l_{k})\Biggr\}. \tag{49} \end{eqnarray}\]

Substituting the values of \(\mu\), \(\sigma\), \(y_{0}\) and \(d\), the OSW (\(l_{h}^{*}\)) and the OSB (\(y_{h}^{*}=y_{h-1}^{*}-l_{h}^{*}\)) are obtained by executing the strata.data() function.

8.9.2 A Numerical Example for Log-Normal Distribution

The ‘hies’ data of size \(N=3566\) is used to demonstrate the application of the stratifyR package on Log-normal population. The ‘hies’ data comes from the HIES survey conducted in Fiji in the year 2010. The data contains only two aspects of the survey, namely Income and Expenditure. In this example, the variable Expenditure is used since it exhibits a 2-parameter Log-normal distribution with the shape and scale parameters as \(meanlog = 9.2804934\) and \(sdlog = 0.6917842\) respectively. The minimum and maximum values are \([y_{0}, y_{L}] = [991.24, 136539.1]\), which implies that \(d=135547.8\).

To construct the OSB (\(h = 2\)) for the Iron data with a fixed total sample size of \(500\), we use the following codes:

data(hies)
Expenditure <- hies$Expenditure
head(Expenditure);length(Expenditure)
#> [1] 10880.58  3633.28  3882.84  4862.67  9237.19  9637.24
#> [1] 3566
hist(Expenditure)

min(Expenditure); max(Expenditure); d=max(Expenditure)-min(Expenditure);d
#> [1] 991.24
#> [1] 136539.1
#> [1] 135547.8
fit <- fitdist(Expenditure, distr="lnorm", method="mle")
fit
#> Fitting of the distribution ' lnorm ' by maximum likelihood 
#> Parameters:
#>          estimate  Std. Error
#> meanlog 9.2804934 0.011584571
#> sdlog   0.6917842 0.008191452
res <- strata.data(Expenditure, h = 2, n=500) 
#> Warning in checkparamlist(arg_startfix$start.arg, arg_startfix$fix.arg, : Some
#> parameter names have no starting/fixed value but have a default value: mean.
#> The program is running, it'll take some time!
summary(res)
#> <div style='margin-bottom:0.5rem'><span style='color:#0b7285;font-weight:700'>Stratification summary</span></div>
#> <div><span style='color:#0b7285;font-weight:600'>Strata (h):</span> 2 &nbsp; <span style='color:#0b7285;font-weight:600'>n:</span> 500 &nbsp; <span style='color:#0b7285;font-weight:600'>N:</span> 3566</div>
#> <div><span style='color:#0b7285;font-weight:600'>Data range:</span> [991.2400, 136539.0600]  (d = 135547.8200)</div>
#> <div><span style='color:#0b7285;font-weight:600'>Best-fit distribution:</span> lnorm</div>
#> 
#> 
#> 
#> Table: Parameter estimate(s)
#> 
#> |Parameter | Estimate|
#> |:---------|--------:|
#> |meanlog   | 9.280493|
#> |sdlog     | 0.691784|

Per-stratum results and totals

Strata	OSB	Wh	Vh	WhSh	nh	Nh	fh
1	17206.65	0.77	14022946	2886.776	214	2749	0.08
2	136539.06	0.23	285033008	3868.016	286	817	0.35
Total		1.00	299055954	6754.792	500	3566	0.14

Similarly, in order to find the OSB and other quantities, we can apply the strata.distr() function to a Log-normal population. Based on the assumption from past knowledge that the population follows Log-normal distribution with the given attributes such as the initial value, distance, parameters, etc., we can execute the following command to obtain the results:

res <- strata.distr(h=2, initval=10, dist=188, distr = "lnorm",
             params = c(meanlog=3.23, sdlog=0.65), n=500, N=1588)
#> The program is running, it'll take some time!
summary(res)
#> <div style='margin-bottom:0.5rem'><span style='color:#0b7285;font-weight:700'>Stratification summary</span></div>
#> <div><span style='color:#0b7285;font-weight:600'>Strata (h):</span> 2 &nbsp; <span style='color:#0b7285;font-weight:600'>n:</span> 500 &nbsp; <span style='color:#0b7285;font-weight:600'>N:</span> 1588</div>
#> <div><span style='color:#0b7285;font-weight:600'>Data range:</span> [10.0000, 198.0000]  (d = 188.0000)</div>
#> <div><span style='color:#0b7285;font-weight:600'>Best-fit distribution:</span> lnorm</div>
#> 
#> 
#> 
#> Table: Parameter estimate(s)
#> 
#> |Parameter | Estimate|
#> |:---------|--------:|
#> |meanlog   |     3.23|
#> |sdlog     |     0.65|

Per-stratum results and totals

Strata	OSB	Wh	Vh	WhSh	nh	Nh	fh
1	39.66	0.76	63.79	5.423	247	1200	0.21
2	198	0.24	516.95	5.535	253	388	0.65
Total		1.00	580.74	10.958	500	1588	0.31

8.10 Stratification for a Survey Variable with Cauchy Distribution

If the study variable follows the Cauchy distribution on the interval \([y_{0}, y_{L}]\), its two-parameter probability density function is given by:

\[\begin{eqnarray} f(y; \mu, \sigma)=\dfrac{1}{{\pi}\sigma\left[1+\left(\dfrac{y-\mu}{\sigma}\right)^2\right]} \;\;\;\;-\infty < y < +\infty \tag{50} \end{eqnarray}\]

where \(\mu\) is the location parameter which specifies the location of the peak of the distribution and \(\sigma\) is the scale parameter, which specifies the half-width at half maximum of the distribution.

8.10.1 DP Solution for Cauchy Distribution

To solve the MPP formulated for Cauchy distribution (50), we apply the algorithm using the DP technique discussed in Section 4. The recurrence relations used to determine the OSB are given by:

For the first stage, \(k=1\), at \(l_{1}^{*}=d_{1}\):

\[\begin{eqnarray} \Phi_{1}d_{1}&=& \textrm{Sqrt}\Biggl\{\dfrac{1}{\pi^{2}}\left[\tan^{-1}\left(\dfrac{d_{1}+y_{0}-\mu}{\sigma}\right)-\tan^{-1}\left(\dfrac{y_{0}-\mu}{\sigma}\right)\right]\nonumber\\[10pt] &&\times\left[\mu\sigma\ln\left(\left(d_{1}+y_{0}-\mu\right)^2+\sigma^2\right)+\left(\mu^2-\sigma^2\right)\tan^{-1}\left(\dfrac{d_{1}+y_{0}-\mu}{\sigma}\right)\right.\nonumber\\[10pt] &&+\sigma\left(d_{1}+y_{0}\right)-\mu\sigma\ln\left(\left(y_{0}-\mu\right)^2+\sigma^2\right)\nonumber\\[10pt] &&-\left.\left(\mu^2-\sigma^2\right)\tan^{-1}\left(\dfrac{y_{0}-\mu}{\sigma}\right)-\sigma y_{0}\right]\nonumber\\[10pt] &&-\left[\dfrac{1}{4\pi^{2}}\left[\sigma\ln\left(\left(d_{1}+y_{0}-\mu\right)^2+\sigma^2\right)+2\mu\tan^{-1}\left(\dfrac{d_{1}+y_{0}-\mu}{\sigma}\right)\right.\right.\nonumber\\[10pt] &&-\left.\left.\sigma\ln\left(\left(y_{0}-\mu\right)^2+\sigma^2\right)-2\mu\tan^{-1}\left(\dfrac{y_{0}-\mu}{\sigma}\right)\right]\right]^{2}\Biggr\} \tag{51} \end{eqnarray}\]

And for the stages \(k\geq2\):

\[\begin{eqnarray} \Phi_{k}d_{k}&=&\min_{0\leq l_{k}\leq d_{k}}\Biggl\{ \textrm{Sqrt}\Biggl\{ \dfrac{1}{\pi^{2}}\left[\tan^{-1}\left(\dfrac{d_{k}+y_{0}-\mu}{\sigma}\right)-\tan^{-1}\left(\dfrac{d_{k}-l_{k}+y_{0}-\mu}{\sigma}\right)\right]\nonumber\\[10pt] &&\times\left[\mu\sigma\ln\left(\left(d_{k}+y_{0}-\mu\right)^2+\sigma^2\right)+\left(\mu^2-\sigma^2\right)\tan^{-1}\left(\dfrac{d_{k}+y_{0}-\mu}{\sigma}\right)\right.\nonumber\\[10pt] &&+\sigma\left(d_{k}+y_{0}\right)-\mu\sigma\ln\left(\left(d_{k}-l_{k}+y_{0}-\mu\right)^2+\sigma^2\right)\nonumber\\[10pt] &&-\left.\left(\mu^2-\sigma^2\right)\tan^{-1}\left(\dfrac{d_{k}-l_{k}+y_{0}-\mu}{\sigma}\right)-\sigma y_{h-1}\right]\nonumber\\[10pt] &&-\left[\dfrac{1}{4\pi^{2}}\left[\sigma\ln\left(\left(d_{k}+y_{0}-\mu\right)^2+\sigma^2\right)+2\mu\tan^{-1}\left(\dfrac{d_{k}+y_{0}-\mu}{\sigma}\right)\right.\right.\nonumber\\[10pt] &&-\left.\left.\sigma\ln\left(\left(d_{k}-l_{k}+y_{0}-\mu\right)^2+\sigma^2\right)-2\mu\tan^{-1}\left(\dfrac{d_{k}-l_{k}+y_{0}-\mu}{\sigma}\right)\right]\right]^{2} \Biggr\} + \Phi_{k-1}(d_{k}-l_{k})\Biggr\}. \tag{52} \end{eqnarray}\]

Substituting the values of \(\mu\), \(\sigma\), \(y_{0}\) and \(d\), the OSW (\(l_{h}^{*}\)) and the OSB (\(y_{h}^{*}=y_{h-1}^{*}-l_{h}^{*}\)) are obtained by executing the strata.data() function.

8.10.2 A Numerical Example for Cauchy Distribution

The Boston data from the MASS package is used to demonstrate the application for Cauchy population. In this example, the variable ‘black’ is used since it exhibits a 2-parameter Cauchy distribution with the location and scale parameters as \(location = 393.864307\) and \(scale = 4.710457\) respectively. The minimum and maximum values are \([y_{0}, y_{L}] = [0.32, 396.9]\), which implies that \(d=396.58\).

data(Boston) #Housing Values in Suburbs of Boston
black = Boston$black
hist(black)

min(black); max(black); d=max(black)-min(black);d
#> [1] 0.32
#> [1] 396.9
#> [1] 396.58
fit <- fitdist(black, distr="cauchy", method="mle")
fit
#> Fitting of the distribution ' cauchy ' by maximum likelihood 
#> Parameters:
#>            estimate Std. Error
#> location 393.863075  0.3060843
#> scale      4.709346  0.3395534
res <- strata.data(black, h = 2, n=500)
#> Warning in checkparamlist(arg_startfix$start.arg, arg_startfix$fix.arg, : Some
#> parameter names have no starting/fixed value but have a default value: mean.
#> The program is running, it'll take some time!
summary(res)
#> <div style='margin-bottom:0.5rem'><span style='color:#0b7285;font-weight:700'>Stratification summary</span></div>
#> <div><span style='color:#0b7285;font-weight:600'>Strata (h):</span> 2 &nbsp; <span style='color:#0b7285;font-weight:600'>n:</span> 500 &nbsp; <span style='color:#0b7285;font-weight:600'>N:</span> 506</div>
#> <div><span style='color:#0b7285;font-weight:600'>Data range:</span> [0.3200, 396.9000]  (d = 396.5800)</div>
#> <div><span style='color:#0b7285;font-weight:600'>Best-fit distribution:</span> cauchy</div>
#> 
#> 
#> 
#> Table: Parameter estimate(s)
#> 
#> |Parameter |   Estimate|
#> |:---------|----------:|
#> |location  | 393.864327|
#> |scale     |   4.708422|

Per-stratum results and totals

Strata	OSB	Wh	Vh	WhSh	nh	Nh	fh
1	364.29	0.2	18335.38	27.028	101	101	1.00
2	396.9	0.8	58.14	6.103	399	405	0.99
Total		1.0	18393.52	33.131	500	506	0.99

Please note that for Cauchy distributions, one might get the error messages “simpleError in optim()…”. You can ignore this error message which simply indicates that when the data is fitted to all ten distributions, it gives errors while computing the parameters of those distributions which are not defined in the negative region. This happens with Cauchy distribution because it is defined on both positive and negative regions.

Similarly, in order to find the OSB and other quantities, we can apply the strata.distr() function to a Cauchy population. Based on the assumption from past knowledge that the population follows Cauchy distribution with the given attributes such as the initial value, distance, parameters, etc., we can execute the following command to obtain the results:

#for a cauchy distribution with initial value of x0=-1, d=2 and 
#location and scale parameters 0 and 1 respectively
res <- strata.distr(h=2, initval=-1, dist=2, distr = "cauchy",
             params = c(location=0, scale=1), n=500, N=5000)
#> The program is running, it'll take some time!
summary(res)
#> <div style='margin-bottom:0.5rem'><span style='color:#0b7285;font-weight:700'>Stratification summary</span></div>
#> <div><span style='color:#0b7285;font-weight:600'>Strata (h):</span> 2 &nbsp; <span style='color:#0b7285;font-weight:600'>n:</span> 500 &nbsp; <span style='color:#0b7285;font-weight:600'>N:</span> 5000</div>
#> <div><span style='color:#0b7285;font-weight:600'>Data range:</span> [-1.0000, 1.0000]  (d = 2.0000)</div>
#> <div><span style='color:#0b7285;font-weight:600'>Best-fit distribution:</span> cauchy</div>
#> 
#> 
#> 
#> Table: Parameter estimate(s)
#> 
#> |Parameter | Estimate|
#> |:---------|--------:|
#> |location  |        0|
#> |scale     |        1|

Per-stratum results and totals

Strata	OSB	Wh	Vh	WhSh	nh	Nh	fh
1	0	0.5	0.08	0.07	250	2500	0.1
2	1	0.5	0.08	0.07	250	2500	0.1
Total		1.0	0.16	0.14	500	5000	0.1

9 Summary of Method Comparison Across All Supported Distributions

The three optimisation methods available in stratifyR — Dynamic Programming ("dp"), COBYLA ("cobyla"), and DIRECT-L + COBYLA ("global") — produce the same optimal stratum boundaries and Neyman cost (\(\sum W_h S_h\)) in many cases, but can diverge meaningfully depending on the shape of the underlying distribution. To illustrate this, we applied all three methods to strata.data() with \(h = 2\) (or \(h = 3\) for Weibull, as in the earlier section) across all ten distributions supported by the package. The results are summarised in Table 1 below.

Table 1: Comparison of DP, COBYLA and GLOBAL methods across all ten distributions supported by stratifyR. WhShTot values shown for h=2 (h=3 for Weibull). Lower WhShTot indicates a more efficient stratification.

Distribution	Example Data	DP	COBYLA	GLOBAL	Outcome
Pareto	Simulated (N=5000, shape=5, scale=8)	0.938	1.204	1.204	DP clearly better
Exponential	Simulated (N=5000, rate=1.36)	0.399	0.425	0.425	DP clearly better
Gamma	Simulated (N=500, shape=3, rate=0.7)	1.347	1.405	1.405	DP clearly better
Log-Normal	HIES Expenditure (N=3566)	6754.790	6801.420	6801.420	DP clearly better
Uniform	Simulated (N=5000, min=2, max=15)	1.880	1.880	1.880	All agree
Normal	Simulated (N=5000, mean=6, sd=2.1)	1.006	1.008	1.008	All agree
Right-Triangular	Simulated (N=1000, a=2, b=8)	0.727	0.727	0.727	All agree
Weibull	Anaemia Iron (N=724, h=3)	2.461	2.433	2.433	COBYLA/GLOBAL marginally better
Triangular	Math final marks (N=354)	10.741	10.715	10.715	COBYLA/GLOBAL marginally better
Cauchy	Simulated (N=3000, location=15, scale=4)	8.442	8.468	8.468	DP marginally better

Three distinct patterns emerge from Table 1. For heavy-tailed, right-skewed distributions — Pareto, Exponential, Gamma and Log-Normal — DP substantially outperforms COBYLA and GLOBAL. The reason is structural: for these distributions the theoretically optimal stratum boundary lies very close to the data minimum, in a region of rapidly changing density. COBYLA’s linear-approximation local search consistently misses this low-boundary optimum, settling instead in a nearby but suboptimal basin. DP avoids this problem entirely because it evaluates the objective on an exhaustive discrete grid and cannot be trapped by a local minimum. The GLOBAL method (DIRECT-L followed by COBYLA refinement) also misses this region in practice, because DIRECT-L’s uniform partition of the search space does not resolve the narrow optimal zone near the lower boundary finely enough before handing off to COBYLA.

For symmetric or bounded distributions — Uniform, Normal and Right-Triangular — all three methods converge to the same solution. The objective landscape for these distributions is smooth and unimodal, so COBYLA’s local search finds the global minimum reliably.

For the Weibull and Triangular distributions, COBYLA and GLOBAL report a marginally lower empirical \(\sum W_h S_h\) than DP. As explained in Section 6.2, this does not mean DP is inferior: the two sets of methods are minimising different objectives. DP minimises the theoretical \(W_h S_h\) derived analytically from the fitted distributional model (via data.root), whereas COBYLA and GLOBAL directly minimise the empirical \(W_h S_h\) from the raw data. The reported WhShTot is always the empirical value, so methods optimising that quantity directly will tend to report a slightly lower number. For these distributions the optimal boundary falls in an interior region of the data range — accessible to COBYLA’s empirical search — so the two approaches reach similar, though not identical, solutions. The DP boundaries remain theoretically justified and generalisable to the underlying population.

For the Cauchy distribution, DP is marginally superior. The Cauchy’s extremely heavy tails and undefined mean make empirical optimisation unstable, whereas DP’s grid-based search over the fitted Cauchy recurrence relations is unaffected by the distributional anomalies.

Overall, DP is the recommended and default method for strata.data(), particularly when the survey variable is right-skewed or heavy-tailed. COBYLA and GLOBAL are provided as fast alternatives and can be useful for preliminary exploration or large datasets where computation time is a concern, but they should be used with caution for distributions in the first group above.

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