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Generalized Process Capability Indices for Progressive Type-II Censored Data

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

2026-07-31

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 1.5975  1.0983 3.7072
#> scale scale   3.5655 0.6925 -0.2083 0.5134
#> 
#> Capability Indices:
#>        item estimate     se    bias    mse
#> Cpy     Cpy   0.9385 0.0794  0.0068 0.0062
#> Cp       Cp   0.6460 0.5900  0.3599 0.4707
#> Cpk     Cpk   0.5874 0.3725  0.1784 0.1678
#> Cpm     Cpm   0.6362 0.2897  0.1288 0.0988
#> CpTk   CpTk   0.8667 0.2637 -0.2422 0.1268
#> Spmk   Spmk   0.6080 0.2954  0.0817 0.0922
#> CNpmc CNpmc   0.5227 0.1332  0.0361 0.0187
#> 
#> 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% 0.7663
#> 2       GPCI   Cpy   0.9385 percentile     parametric  0.05        95% 0.7498
#> 3       GPCI   Cpy   0.9385 percentile     parametric  0.01        99% 0.7075
#> 4       GPCI    Cp   0.6460 percentile     parametric  0.10        90% 0.3850
#> 5       GPCI    Cp   0.6460 percentile     parametric  0.05        95% 0.3449
#> 6       GPCI    Cp   0.6460 percentile     parametric  0.01        99% 0.2684
#> 7       GPCI   Cpk   0.5874 percentile     parametric  0.10        90% 0.2954
#> 8       GPCI   Cpk   0.5874 percentile     parametric  0.05        95% 0.2648
#> 9       GPCI   Cpk   0.5874 percentile     parametric  0.01        99% 0.2186
#> 10      GPCI   Cpm   0.6362 percentile     parametric  0.10        90% 0.3791
#> 11      GPCI   Cpm   0.6362 percentile     parametric  0.05        95% 0.3385
#> 12      GPCI   Cpm   0.6362 percentile     parametric  0.01        99% 0.2651
#> 13      GPCI  CpTk   0.8667 percentile     parametric  0.10        90% 0.1242
#> 14      GPCI  CpTk   0.8667 percentile     parametric  0.05        95% 0.0272
#> 15      GPCI  CpTk   0.8667 percentile     parametric  0.01        99% 0.0000
#> 16      GPCI  Spmk   0.6080 percentile     parametric  0.10        90% 0.3232
#> 17      GPCI  Spmk   0.6080 percentile     parametric  0.05        95% 0.3055
#> 18      GPCI  Spmk   0.6080 percentile     parametric  0.01        99% 0.1816
#> 19      GPCI CNpmc   0.5227 percentile     parametric  0.10        90% 0.3501
#> 20      GPCI CNpmc   0.5227 percentile     parametric  0.05        95% 0.3175
#> 21      GPCI CNpmc   0.5227 percentile     parametric  0.01        99% 0.2541
#> 22 Parameter shape   2.1242 percentile     parametric  0.10        90% 1.4309
#> 23 Parameter shape   2.1242 percentile     parametric  0.05        95% 1.2899
#> 24 Parameter shape   2.1242 percentile     parametric  0.01        99% 1.1224
#> 25 Parameter scale   3.5655 percentile     parametric  0.10        90% 2.3337
#> 26 Parameter scale   3.5655 percentile     parametric  0.05        95% 2.2808
#> 27 Parameter scale   3.5655 percentile     parametric  0.01        99% 1.6561
#>     upper  width
#> 1  1.0027 0.2364
#> 2  1.0027 0.2529
#> 3  1.0027 0.2952
#> 4  1.7206 1.3356
#> 5  2.8879 2.5430
#> 6  3.2836 3.0152
#> 7  1.3591 1.0637
#> 8  1.4415 1.1767
#> 9  1.8864 1.6679
#> 10 1.2861 0.9071
#> 11 1.3115 0.9731
#> 12 1.3552 1.0901
#> 13 0.9513 0.8271
#> 14 0.9923 0.9651
#> 15 0.9986 0.9986
#> 16 1.2472 0.9241
#> 17 1.2559 0.9504
#> 18 1.3891 1.2075
#> 19 0.7464 0.3963
#> 20 0.7513 0.4339
#> 21 0.7592 0.5051
#> 22 5.8099 4.3789
#> 23 6.0860 4.7961
#> 24 8.5446 7.4222
#> 25 4.4129 2.0792
#> 26 4.9230 2.6422
#> 27 5.2383 3.5822

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