Lindley Approximation Method for Generalized Process Capability Indices under Progressive Type-II Censoring

Shikhar Tyagi, Sumit Kumar, Arvind Pandey, Bhupendra Singh, Vrijesh Tripathi

Introduction

The gpciLindApproxProgII package provides a Bayesian statistical framework for computing Generalized Process Capability Indices (GPCIs) under Progressive Type-II Censored Data using Lindley’s 3rd-order Approximation Method.

Progressive Type-II Censoring Model

Let \(n\) units be placed on test and \(m\) failure times \(X = (x_1, x_2, \dots, x_m)\) be observed under progressive removal scheme \(R = (R_1, R_2, \dots, R_m)\). The progressive log-likelihood function is:

\[\ell(\theta) = \sum_{i=1}^m \log f(x_i; \theta) + \sum_{i=1}^m R_i \log S(x_i; \theta)\]

Example Analysis

Below is a demonstration of fitting progressive Type-II failure data with custom user functions or built-in distributions.

# Failure times and progressive removal scheme
x <- c(0.5, 1.2, 2.1, 3.4, 4.8)
r <- c(1, 0, 2, 0, 1)

# Fit model using Lindley approximation and chain generation
fit <- lindley_prog_gpci(
  x = x,
  r_removals = r,
  distribution = dist_weibull(),
  USL = 6, LSL = 0,
  chain_length = 200,
  burn_in = 50,
  thinning = 1,
  B = 50
)

# Print Summary Table
summary(fit)
#>     Index MLE_Estimate Lindley_Estimate    Chain_Mean          Bias
#> 1     Cpy 4.953015e-01     4.247116e-01  4.177683e-01 -7.753318e-02
#> 2      Cp 1.000000e+04     1.000000e+04  7.730188e+03 -2.269812e+03
#> 3     Cpk 2.977036e-73    -3.832680e-43  1.404065e-03  1.404065e-03
#> 4     Cpu 2.000000e+04     2.000000e+04  1.546033e+04 -4.539668e+03
#> 5     Cpl 2.977036e-73    -3.832680e-43  4.401479e-02  4.401479e-02
#> 6     Cpm 3.333333e-01     3.333333e-01  2.880503e-01 -4.528308e-02
#> 7    Cpmk 9.923452e-78    -1.277560e-47  1.366696e-03  1.366696e-03
#> 8    CpTk 1.249103e-01     1.102875e-01  1.000801e-01 -2.483017e-02
#> 9    Spmk 2.141650e+03     1.977308e+03  1.231935e+03 -9.097149e+02
#> 10    Cpc 1.000000e+04     1.000000e+04  7.732466e+03 -2.267534e+03
#> 11   CNpk 0.000000e+00     0.000000e+00 -7.214422e-03 -7.214422e-03
#> 12  CNpmc 4.792753e-02     4.136523e-02  3.720504e-02 -1.072249e-02
#> 13 CNpmkc 0.000000e+00     0.000000e+00  2.266697e-04  2.266697e-04
#> 14  Cp_uv 0.000000e+00     0.000000e+00  1.366696e-03  1.366696e-03
#>             MSE   Risk_Value   HPD90_Lower  HPD90_Upper   HPD95_Lower
#> 1  2.343364e-02 2.936810e-02  2.601978e-01 5.749860e-01  2.432379e-01
#> 2  2.232023e+07 2.232930e+07  1.404483e-01 1.000000e+04  7.520631e-02
#> 3  7.029974e-04 7.049693e-04 -7.174927e-02 1.590372e-06 -7.174927e-02
#> 4  8.928267e+07 8.930082e+07  2.758702e-02 2.000000e+04 -4.174774e-02
#> 5  1.263764e-02 1.458923e-02  0.000000e+00 2.635498e-01  0.000000e+00
#> 6  1.030821e-02 1.234337e-02  1.294286e-01 3.333333e-01  7.041530e-02
#> 7  6.372686e-04 6.391369e-04 -6.771775e-02 1.819564e-08 -6.771775e-02
#> 8  1.385563e-03 1.999557e-03  5.899898e-02 1.440558e-01  3.837450e-02
#> 9  1.493858e+06 1.497489e+06  4.218943e-05 2.301085e+03  4.218943e-05
#> 10 2.229910e+07 2.230816e+07  1.605108e-01 1.000000e+04  7.041530e-02
#> 11 7.717603e-03 7.769588e-03  0.000000e+00 1.564175e-01 -1.310207e-01
#> 12 3.026810e-04 4.174476e-04  1.428994e-02 5.518352e-02  7.599846e-03
#> 13 1.715619e-06 1.767000e-06 -1.546218e-03 2.814646e-04 -1.546218e-03
#> 14 6.372686e-04 6.391369e-04 -6.771775e-02 1.819564e-08 -6.771775e-02
#>     HPD95_Upper   HPD99_Lower  HPD99_Upper    HW_Stat HW_Pvalue HW_Passed
#> 1  7.088777e-01  1.946527e-01 9.038548e-01 0.15621723       0.5      TRUE
#> 2  1.000000e+04  4.012932e-04 1.000000e+04 0.06293788       0.5      TRUE
#> 3  5.439582e-02 -7.174927e-02 1.213582e-01 0.21609261       0.5      TRUE
#> 4  2.000000e+04 -6.991147e-02 2.000000e+04 0.06293746       0.5      TRUE
#> 5  3.159464e-01  0.000000e+00 3.543571e-01 0.05472611       0.5      TRUE
#> 6  3.333333e-01  3.981231e-04 3.333333e-01 0.10574099       0.5      TRUE
#> 7  5.088988e-02 -6.771775e-02 1.171238e-01 0.21550619       0.5      TRUE
#> 8  1.440558e-01  1.045417e-02 1.477033e-01 0.06443744       0.5      TRUE
#> 9  2.435750e+03  4.218943e-05 2.689890e+03 0.07940454       0.5      TRUE
#> 10 1.000000e+04  3.981223e-04 1.000000e+04 0.06333397       0.5      TRUE
#> 11 2.205716e-01 -4.669647e-01 2.205716e-01 0.16040564       0.5      TRUE
#> 12 5.539170e-02  1.246212e-05 5.518352e-02 0.12133421       0.5      TRUE
#> 13 3.443590e-03 -1.546218e-03 7.548219e-03 0.15528746       0.5      TRUE
#> 14 5.088988e-02 -6.771775e-02 1.171238e-01 0.21550619       0.5      TRUE
#>    Convergence_Prob Boot95_Lower Boot95_Upper
#> 1               0.5 3.275979e-01 1.584475e+00
#> 2               0.5 1.966687e+03 1.000000e+04
#> 3               0.5 0.000000e+00 1.675544e-02
#> 4               0.5 3.933153e+03 2.000000e+04
#> 5               0.5 0.000000e+00 2.215312e-01
#> 6               0.5 2.460495e-01 3.333333e-01
#> 7               0.5 0.000000e+00 1.534837e-02
#> 8               0.5 3.524268e-02 1.552800e-01
#> 9               0.5 4.989649e-01 2.669147e+03
#> 10              0.5 2.022567e+03 1.000000e+04
#> 11              0.5 0.000000e+00 3.581466e-02
#> 12              0.5 2.915852e-02 5.555556e-02
#> 13              0.5 0.000000e+00 9.138423e-04
#> 14              0.5 0.000000e+00 1.534837e-02

Plotting Results

plot(fit)