Working with Order Statistics in the mos Package

Introduction

The mos package provides a suite of tools to work with order statistics, including:

• Generating random values of order statistics and censored/record samples,
• Analytical computation of moments (mean, variance, skewness, kurtosis),
• Simulation-based estimation of moments for some of continuous distributions.

These tools are useful for researchers in statistics, reliability, and simulation studies.

Generating Order Statistics

You can simulate order statistics from continuous distributions using the ros() function. This function accepts either:

• A named distribution available in base R (e.g., “norm”, “exp”), or
• A user-defined quantile function using the qf argument.

ros(5, r = 2, n = 10, dist = "norm", mean = 0, sd = 1)
#> [1] -1.072941 -1.514401 -2.151302 -1.114095 -1.160008

qgumbel <- function(p, mu = 0, beta = 1) mu - beta * log(-log(p))
ros(5, r = 3, n = 15, qf = "qgumbel", mu = 0, beta = 1)
#> [1] -0.5536969 -0.6472786 -0.3054830 -0.4340630 -0.7906341

ros(3, 1:5, 5, dist = "unif", min = 1, max = 5)
#>            r1       r2       r3       r4       r5
#> [1,] 2.732895 3.152392 3.458109 4.788939 4.999965
#> [2,] 1.058484 2.108461 2.692568 2.955375 3.435296
#> [3,] 2.379856 3.149699 3.490129 4.195683 4.877305

Moments of Order Statistics

The package provides two categories of moment computation:

• Analytical formulas (closed-form), where available
• Simulation-based estimation otherwise

mo_unif(r = 2, n = 5, k = 1)  # Mean of 2nd order statistic
#> [1] 0.3333333

mo_exp(r = 3, n = 6, k = 2, mu = 0, sigma = 1)
#> [1] 0.5105556

mo_norm(r = 2, n = 10, k = 1)
#> [1] -1.003343

mo_beta(r = 2, n = 5, k = 2, a = 2, b = 3)
#> [1] 0.09487993

Censored Samples

The mos package supports simulation of censored data using several standard censoring schemes:

rcens(n = 10, dist = "exp", type = "I", cens.time = 2, rate = 1)
#>  [1] 0.04965609 1.67364631 1.25681359 0.07160058 0.42816672 0.19903825
#>  [7] 0.02879249 1.12775815 0.20484918 0.32955892

rcens(n = 10, r = 5, dist = "norm", type = "II", mean = 0, sd = 1)
#> [1] -0.47197938 -0.37010037 -0.08041435 -0.05771255  0.02607693

rpcens2(n = 10, R = c(2, 1, 2, 0, 0), dist = "exp", rate = 1)
#> [1] 0.02910825 0.15546476 0.31029202 0.47766020 0.86184076

Record Values

The rkrec() function generates upper and lower k-records from continuous distributions.

rkrec(size = 5, k = 2, record = "upper", dist = "norm", mean = 0, sd = 1)
#> [1] -0.1252028 -1.2616253 -0.5911243 -1.6933552  0.2620640

rkrec(size = 5, k = 3, record = "lower", dist = "exp", rate = 1)
#> [1] 0.3086123 1.9005258 0.8497093 2.1867893 0.8350877

An Application of the Package in Graphical Goodness-of-Fit

This section compares analytical and simulated values of the second moment, variance, skewness, and kurtosis of order statistics from the Uniform(0,1) distribution:

if (requireNamespace("moments", quietly = TRUE)) {
  library(moments)
  set.seed(123)
  x <- ros(1e4, r = 1:25, n = 25, dist = "unif")
  observed <- colMeans(x^2)
  expected <- mo_unif(r = 1:25, n = 25, k = 2, a = 0, b = 1)
  oldpar <- par(mfrow = c(2, 2))
  on.exit(par(oldpar))
  plot(observed, expected, main = "2nd Moment of U(0,1)", xlab = "Observed", ylab = "Expected")
  abline(0, 1, col = 2)

  var_obs <- apply(x, 2, var)
  var_expc <- varOS(1:25, 25, dist = "unif", a = 0, b = 1)
  plot(var_obs, var_expc, main = "Variance of Order Statistics from U(0,1)",
       xlab = "Observed", ylab = "Expected")
  abline(0, 1, col = 2)

  skew_obs <- apply(x, 2, moments::skewness)
  skew_expc <- skewOS(1:25, 25, dist = "unif", a = 0, b = 1)
  plot(skew_obs, skew_expc, main = "Skewness of Order Statistics from U(0,1)",
       xlab = "Observed", ylab = "Expected")
  abline(0, 1, col = 2)

  kurt_obs <- apply(x, 2, moments::kurtosis)
  kurt_expc <- kurtOS(1:25, 25, dist = "unif", a = 0, b = 1)
  plot(kurt_obs, kurt_expc, main = "Kurtosis of Order Statistics from U(0,1)",
       xlab = "Observed", ylab = "Expected")
  abline(0, 1, col = 2)
}

The plots show a strong alignment between the observed and expected values along the identity line. This suggests that the order statistics simulated from the Uniform(0,1) distribution closely follow their theoretical moments. Such alignment in graphical comparisons is often taken as visual evidence of a good fit between observed data and the theoretical model.

Conclusion

The mos package integrates analytical and simulation-based methods for working with order statistics, censored data, and record values. It serves as a powerful and flexible toolkit for statisticians, particularly those involved in reliability analysis and the study of ordered data.