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MCMC Estimation of Generalized Process Capability Indices under Hybrid Type-II Censoring

Shikhar Tyagi, Vrijesh Tripathi

Introduction

The gpcihybridIImcmc package provides Bayesian Markov Chain Monte Carlo (MCMC) estimation methods using Metropolis-Hastings within Gibbs sampler for Generalized Process Capability Indices (GPCIs) under Hybrid Type-II censored lifetime data.

Under Hybrid Type-II censoring (Epstein 1954; Childs et al. 2003; Kundu and Pradhan 2009), \(n\) identical units are placed on life testing. The experiment stops at \(T^* = \max(x_r, T_c)\), where \(r \le n\) is the target number of failures and \(T_c > 0\) is the pre-fixed censoring time.

Supported capability indices include \(C_{py}\), \(C_p\), \(C_{pk}\), \(C_{pu}\), \(C_{pl}\), \(C_{pm}\), \(C_{pmk}\), \(S_{pmk}\), \(C_{pTk}\), \(C_{pc}\), \(C_{Np}\), \(C_{Npk}\), \(C_{Npm}\), \(C_{Npmk}\), \(C_{Npmc}\), and \(C_{Npmkc}\).

Usage with Custom Probability Functions

Users can pass custom probability density/mass functions (pdf), cumulative distribution functions (cdf), and survival functions (surv) as R functions:

library(gpcihybridIImcmc)

# User-defined Exponential lifetime distribution functions
my_pdf  <- function(x, rate) stats::dexp(x, rate = rate)
my_cdf  <- function(q, rate) stats::pexp(q, rate = rate)
my_surv <- function(q, rate) stats::pexp(q, rate = rate, lower.tail = FALSE)

# Hybrid Type-II censored sample: n = 10 units, target r = 3, censoring time tc = 2.5
data <- c(0.5, 1.2, 2.1, 3.4)

fit <- gpci_hybrid2_mcmc(
  x = data,
  r = 3,
  tc = 2.5,
  n = 10,
  pdf = my_pdf,
  cdf = my_cdf,
  surv = my_surv,
  param_names = "rate",
  start = c(rate = 0.5),
  length_chain = 1000,
  burn_in = 200,
  thinning = 2,
  USL = 8,
  LSL = 0,
  target = 4
)

print(fit)
#> --- Bayesian MCMC Estimation of GPCIs under Hybrid Type-II Censoring ---
#> Distribution:  Custom 
#> Observed Failures (d):  4  | Target (r):  3  | Total (n):  10 
#> Censoring Time (Tc):  2.5  | Effective Termination (T*):  2.5 
#> Specification Limits: LSL = 0 | Target = 4 | USL = 8 
#> 
#> Initial Parameter MLEs (under Hybrid Type-II Censoring):
#>   rate 
#> 0.1803 
#> 
#> Point Estimates of GPCIs (at Initial Estimates):
#>    Cpy     Cp    Cpk    Cpm   Cpmk   Spmk  CNpmc 
#> 0.7657 0.2404 0.1474 0.2315 0.1420 0.3802 0.2154 
#> 
#> Posterior GPCI Summary (Thinned MCMC Chain):
#>   Variable Estimate    Bias    MSE Bayes_Risk HPD_95_Lower HPD_95_Upper
#> 1      Cpy   0.7561 -0.0095 0.0219     0.0219       0.4942       0.9483
#> 2       Cp   0.2634  0.0230 0.0109     0.0104       0.1120       0.4766
#> 3      Cpk   0.1590  0.0116 0.0245     0.0245      -0.1070       0.3333
#> 4      Cpm   0.2525  0.0210 0.0112     0.0108       0.0933       0.4379
#> 5     Cpmk   0.1577  0.0157 0.0222     0.0221      -0.0893       0.3333
#> 6     Spmk   0.3932  0.0131 0.0161     0.0160       0.1845       0.5835
#> 7    CNpmc   0.2237  0.0084 0.0059     0.0059       0.0965       0.3473
#>   HW_Passed
#> 1      TRUE
#> 2      TRUE
#> 3      TRUE
#> 4      TRUE
#> 5      TRUE
#> 6      TRUE
#> 7      TRUE

Statistical Summaries & Convergence Diagnostics

The package automatically calculates: - Point estimates and initial MLE estimates under Hybrid Type-II censoring. - Posterior mean estimates, bias, Mean Squared Error (MSE), and Bayes Risk under squared error loss. - Highest Posterior Density (HPD) credible intervals at 90%, 95%, and 99% levels. - Heidelberger and Welch’s MCMC Convergence Diagnostics (stationarity and half-width tests). - Empirical coverage probabilities.

References

These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.