Lindley Approximation Method for Generalized Process Capability Indices under Progressive Type-II Censoring
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 4.319322e-01 -6.336927e-02
#> 2 Cp 1.000000e+04 1.000000e+04 7.943009e+03 -2.056991e+03
#> 3 Cpk 2.977036e-73 -3.832680e-43 -2.253558e+04 -2.253558e+04
#> 4 Cpu 2.000000e+04 2.000000e+04 -6.749615e+03 -2.674962e+04
#> 5 Cpl 2.977036e-73 -3.832680e-43 2.263563e+04 2.263563e+04
#> 6 Cpm 3.333333e-01 3.333333e-01 2.905193e-01 -4.281401e-02
#> 7 Cpmk 9.923452e-78 -1.277560e-47 6.294600e-04 6.294600e-04
#> 8 CpTk 1.249103e-01 1.102875e-01 9.619533e-02 -2.871494e-02
#> 9 Spmk 2.141650e+03 1.977307e+03 1.205665e+03 -9.359847e+02
#> 10 Cpc 1.000000e+04 1.000000e+04 7.898492e+03 -2.101508e+03
#> 11 CNpk 0.000000e+00 0.000000e+00 -2.496575e-03 -2.496575e-03
#> 12 CNpmc 4.792753e-02 4.136523e-02 3.727376e-02 -1.065377e-02
#> 13 CNpmkc 0.000000e+00 0.000000e+00 2.645444e-04 2.645444e-04
#> 14 Cp_uv 0.000000e+00 0.000000e+00 6.294600e-04 6.294600e-04
#> MSE Risk_Value HPD90_Lower HPD90_Upper HPD95_Lower
#> 1 3.958480e-02 4.355839e-02 2.497049e-01 6.979403e-01 2.457917e-01
#> 2 2.016490e+07 2.017312e+07 1.489648e-01 1.000000e+04 1.084682e-01
#> 3 1.015705e+11 1.015706e+11 -1.778083e-02 5.220120e-02 -7.555311e-02
#> 4 1.025546e+11 1.025547e+11 -5.233549e-04 2.000000e+04 -4.565391e-02
#> 5 1.024740e+11 Inf 0.000000e+00 2.713795e-01 0.000000e+00
#> 6 9.706370e-03 1.152640e-02 1.333798e-01 3.333333e-01 9.052583e-02
#> 7 1.617034e-03 1.617430e-03 -1.577846e-02 4.941822e-02 -7.223266e-02
#> 8 1.642190e-03 2.462805e-03 5.530085e-02 1.426485e-01 3.952794e-02
#> 9 1.504394e+06 1.508130e+06 3.936062e-05 2.149340e+03 3.936062e-05
#> 10 2.064314e+07 2.065154e+07 1.375262e-01 1.000000e+04 9.052583e-02
#> 11 5.476034e-03 5.482265e-03 -4.044544e-02 1.050967e-01 -2.057551e-01
#> 12 2.962194e-04 4.095210e-04 1.411590e-02 5.555526e-02 9.868358e-03
#> 13 3.931540e-06 4.001527e-06 -1.723384e-03 5.114271e-09 -1.797293e-03
#> 14 1.617034e-03 1.617430e-03 -1.577846e-02 4.941822e-02 -7.223266e-02
#> HPD95_Upper HPD99_Lower HPD99_Upper HW_Stat HW_Pvalue HW_Passed
#> 1 8.064298e-01 2.111268e-01 1.152145e+00 0.11517442 0.5 TRUE
#> 2 1.000000e+04 2.942473e-02 1.000000e+04 0.11434753 0.5 TRUE
#> 3 4.851088e-02 -8.016170e-02 2.046875e-01 0.08755075 0.5 TRUE
#> 4 2.000000e+04 -7.555311e-02 2.000000e+04 0.08872816 0.5 TRUE
#> 5 2.989924e-01 0.000000e+00 3.367854e-01 0.08755075 0.5 TRUE
#> 6 3.333333e-01 2.694228e-02 3.333333e-01 0.08862509 0.5 TRUE
#> 7 4.216781e-02 -7.223266e-02 2.040991e-01 0.12142416 0.5 TRUE
#> 8 1.455746e-01 2.093363e-02 1.611541e-01 0.10451257 0.5 TRUE
#> 9 2.440702e+03 3.936062e-05 2.769181e+03 0.09656982 0.5 TRUE
#> 10 1.000000e+04 2.694228e-02 1.000000e+04 0.13407097 0.5 TRUE
#> 11 1.104370e-01 -2.550876e-01 3.115162e-01 0.16002632 0.5 TRUE
#> 12 5.555526e-02 2.949662e-03 5.555526e-02 0.06580612 0.5 TRUE
#> 13 2.578627e-03 -3.247496e-03 8.312902e-03 0.17478519 0.5 TRUE
#> 14 4.216781e-02 -7.223266e-02 2.040991e-01 0.12142416 0.5 TRUE
#> Convergence_Prob Boot95_Lower Boot95_Upper
#> 1 0.5 3.822584e-01 1.584475e+00
#> 2 0.5 1.000000e+04 1.000000e+04
#> 3 0.5 0.000000e+00 2.601088e-12
#> 4 0.5 2.000000e+04 2.000000e+04
#> 5 0.5 0.000000e+00 2.601088e-12
#> 6 0.5 3.333333e-01 3.333333e-01
#> 7 0.5 0.000000e+00 8.670292e-17
#> 8 0.5 3.395818e-03 1.529376e-01
#> 9 0.5 4.065782e-02 2.399687e+03
#> 10 0.5 1.000000e+04 1.000000e+04
#> 11 0.5 0.000000e+00 0.000000e+00
#> 12 0.5 3.582793e-02 5.555556e-02
#> 13 0.5 0.000000e+00 0.000000e+00
#> 14 0.5 0.000000e+00 7.648203e-17
