Generalized Process Capability Indices for Progressive Type-II Censored Data

Introduction

The gpciProgTyII package provides a unified statistical framework for evaluating classical and Generalized Process Capability Indices (GPCIs) under Progressive Type-II Censored Data.

Key features include: 1. Parameter estimation for progressive Type-II censored data using the MleCensoR package (mle_progressive_type2). 2. Evaluation of GPCIs including \(C_{py}\), \(S_{pmk}\), \(C_{pTk}\), \(C_{pc}\), \(C_{Npmc}\), \(C_{Npmkc}\), \(C_{Npk}\), and Vännman’s \(C_p(u,v)\) family. 3. Computation of parametric and non-parametric bootstrap confidence intervals at 90%, 95%, and 99% levels of significance. 4. Calculation of Standard Errors (SE), Mean Squared Error (MSE), Bias, and empirical coverage probabilities for model parameters and capability indices.

Example: Progressive Type-II Censored Analysis

# Load distribution and define progressive data
dist_w <- dist_weibull(shape = 1.5, scale = 4.0)

# Observed failure times under progressive censoring
x <- c(0.8, 1.5, 2.3, 3.1, 4.2)
r_scheme <- c(1, 0, 2, 0, 1)

# Fit model parameters and compute capability indices
fit <- capability_prog(
  x = x, r_removals = r_scheme,
  distribution = dist_w,
  USL = 6.0, LSL = 0.5, target = 3.25,
  indices = c("Cpy", "Cp", "Cpk", "Cpm", "CpTk", "Spmk", "CNpmc")
)

print(fit)
#> --- Progressive Type-II GPCI Analysis (Class: gpc_prog_fit) ---
#> Distribution:  Weibull 
#> Parameters:    shape = 2.1242, scale = 3.5655 
#> Spec Limits:  LSL = 0.5 , USL = 6 , Target = 3.25 
#> Mode:          moments 
#> Expected Nonconforming (p_hat):  6.4045 %
#> 
#> Point Estimates of Capability Indices:
#>    Cpy     Cp    Cpk    Cpm   CpTk   Spmk  CNpmc 
#> 0.9385 0.6460 0.5874 0.6362 0.8667 0.6080 0.5227

# Compute Bootstrap Confidence Intervals at 90%, 95%, and 99%
ci <- boot_ci_prog(fit, B = 50, alpha = c(0.10, 0.05, 0.01), method = "percentile")
print(ci)
#> --- Progressive Type-II Bootstrap Confidence Intervals ---
#> Bootstrap Type:    parametric 
#> CI Method:         percentile 
#> Replicates (B):    50 
#> 
#> Performance Summary (Standard Error, Bias, MSE):
#> Parameters:
#>        item estimate     se    bias    mse
#> shape shape   2.1242 1.6787  1.0392 3.8417
#> scale scale   3.5655 0.7467 -0.1028 0.5570
#> 
#> Capability Indices:
#>        item estimate     se    bias    mse
#> Cpy     Cpy   0.9385 0.0711  0.0049 0.0050
#> Cp       Cp   0.6460 0.4964  0.3202 0.3440
#> Cpk     Cpk   0.5874 0.4217  0.1913 0.2108
#> Cpm     Cpm   0.6362 0.3306  0.1504 0.1297
#> CpTk   CpTk   0.8667 0.2768 -0.2147 0.1212
#> Spmk   Spmk   0.6080 0.2858  0.0803 0.0865
#> CNpmc CNpmc   0.5227 0.1190  0.0436 0.0158
#> 
#> Confidence Intervals (90%, 95%, 99%):
#>         type index estimate     method bootstrap_type alpha conf_level  lower
#> 1       GPCI   Cpy   0.9385 percentile     parametric  0.10        90% 0.8065
#> 2       GPCI   Cpy   0.9385 percentile     parametric  0.05        95% 0.7667
#> 3       GPCI   Cpy   0.9385 percentile     parametric  0.01        99% 0.7448
#> 4       GPCI    Cp   0.6460 percentile     parametric  0.10        90% 0.4616
#> 5       GPCI    Cp   0.6460 percentile     parametric  0.05        95% 0.4433
#> 6       GPCI    Cp   0.6460 percentile     parametric  0.01        99% 0.4115
#> 7       GPCI   Cpk   0.5874 percentile     parametric  0.10        90% 0.3425
#> 8       GPCI   Cpk   0.5874 percentile     parametric  0.05        95% 0.3025
#> 9       GPCI   Cpk   0.5874 percentile     parametric  0.01        99% 0.2789
#> 10      GPCI   Cpm   0.6362 percentile     parametric  0.10        90% 0.4464
#> 11      GPCI   Cpm   0.6362 percentile     parametric  0.05        95% 0.4101
#> 12      GPCI   Cpm   0.6362 percentile     parametric  0.01        99% 0.3822
#> 13      GPCI  CpTk   0.8667 percentile     parametric  0.10        90% 0.1173
#> 14      GPCI  CpTk   0.8667 percentile     parametric  0.05        95% 0.0431
#> 15      GPCI  CpTk   0.8667 percentile     parametric  0.01        99% 0.0211
#> 16      GPCI  Spmk   0.6080 percentile     parametric  0.10        90% 0.3578
#> 17      GPCI  Spmk   0.6080 percentile     parametric  0.05        95% 0.3539
#> 18      GPCI  Spmk   0.6080 percentile     parametric  0.01        99% 0.3504
#> 19      GPCI CNpmc   0.5227 percentile     parametric  0.10        90% 0.4012
#> 20      GPCI CNpmc   0.5227 percentile     parametric  0.05        95% 0.3743
#> 21      GPCI CNpmc   0.5227 percentile     parametric  0.01        99% 0.3527
#> 22 Parameter shape   2.1242 percentile     parametric  0.10        90% 1.8421
#> 23 Parameter shape   2.1242 percentile     parametric  0.05        95% 1.7104
#> 24 Parameter shape   2.1242 percentile     parametric  0.01        99% 1.5369
#> 25 Parameter scale   3.5655 percentile     parametric  0.10        90% 2.4097
#> 26 Parameter scale   3.5655 percentile     parametric  0.05        95% 2.2899
#> 27 Parameter scale   3.5655 percentile     parametric  0.01        99% 2.1683
#>     upper  width
#> 1  1.0027 0.1962
#> 2  1.0027 0.2360
#> 3  1.0027 0.2579
#> 4  1.9286 1.4671
#> 5  2.1162 1.6729
#> 6  2.5830 2.1715
#> 7  1.5429 1.2004
#> 8  1.6942 1.3917
#> 9  2.2546 1.9756
#> 10 1.3726 0.9262
#> 11 1.5631 1.1529
#> 12 1.9203 1.5382
#> 13 0.9541 0.8368
#> 14 0.9698 0.9267
#> 15 0.9739 0.9528
#> 16 1.3001 0.9423
#> 17 1.3995 1.0456
#> 18 1.4337 1.0833
#> 19 0.7623 0.3610
#> 20 0.7898 0.4155
#> 21 0.8255 0.4728
#> 22 6.1765 4.3344
#> 23 7.0110 5.3007
#> 24 9.7375 8.2006
#> 25 4.8721 2.4624
#> 26 4.9854 2.6955
#> 27 5.0140 2.8458