Generalized Process Capability Indices for Interval-Censored Data

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

2026-08-04

Introduction

The gpciIntCensor package provides a unified, comprehensive framework for evaluating Generalized Process Capability Indices (GPCIs) under interval-censored data using Maximum Likelihood Estimation (MLE) via MleCensoR and bootstrap confidence intervals.

Supported capability indices include: * \(C_{py}\) (Maiti et al., 2010) * \(S_{pmk}\) (Dey & Saha, 2019) * \(C_{pTk}\) (Saha et al., 2019) * \(C_{pc}\) (Saha et al., 2022) * \(C_{Npmc}\) (Alotaibi et al., 2022) * \(C_{Npmkc}\) (Saha et al., 2024) * \(C_{Npk}\) (Saha et al., 2018) * Vännman’s \(C_p(u,v)\) family and quantile analogs.

Workflow Example

1. Define Distribution and Generate Interval-Censored Data

# Define normal distribution
dist_norm <- dist_normal(mean = 10, sd = 1.5)

# Simulate interval-censored data
set.seed(123)
true_vals <- rnorm(30, mean = 10, sd = 1.5)
data_left <- true_vals - 0.25
data_right <- true_vals + 0.25

2. Fit Parameters via MLE for Interval-Censored Data

dist_fitted <- fit_distribution_censor(data_left, data_right, dist_norm)
print(dist_fitted$params)
#> $mean
#> [1] 9.929349
#> 
#> $sd
#> [1] 1.439576

3. Compute Capability Indices

fit_cap <- capability_censor(
  data_left = data_left,
  data_right = data_right,
  distribution = dist_norm,
  USL = 14,
  LSL = 6,
  target = 10,
  indices = c("Cpy", "Cp", "Cpk", "Cpm", "Cpmk", "Spmk", "CpTk", "CNpmc"),
  mode = "moments"
)

print(fit_cap)
#> --- Process Capability Analysis for Interval-Censored Data ---
#> Distribution:  normal 
#> Parameters:    mean = 9.9293, sd = 1.4396 
#> Spec Limits:  LSL = 6 , USL = 14 , Target = 10 
#> Mode:          moments 
#> Expected Nonconforming (p_hat):  0.5516 %
#> 
#> Point Estimates of Capability Indices:
#>    Cpy     Cp    Cpk    Cpm   Cpmk   Spmk   CpTk  CNpmc 
#> 0.9972 0.9262 0.9098 0.9251 0.9087 0.9240 0.9607 0.7601

4. Bootstrap Confidence Intervals (90%, 95%, 99%)

ci_res <- boot_ci_censor(
  fit = fit_cap,
  B = 100,
  alpha = c(0.10, 0.05, 0.01),
  method = "percentile",
  type = "nonparametric"
)
#> Warning in norm.inter(t, alpha): extreme order statistics used as endpoints
#> Warning in norm.inter(t, alpha): extreme order statistics used as endpoints
#> Warning in norm.inter(t, alpha): extreme order statistics used as endpoints
#> Warning in norm.inter(t, alpha): extreme order statistics used as endpoints
#> Warning in norm.inter(t, alpha): extreme order statistics used as endpoints
#> Warning in norm.inter(t, alpha): extreme order statistics used as endpoints
#> Warning in norm.inter(t, alpha): extreme order statistics used as endpoints
#> Warning in norm.inter(t, alpha): extreme order statistics used as endpoints

print(ci_res)
#> --- Bootstrap Confidence Intervals (Interval-Censored Data) ---
#> Bootstrap Type:    nonparametric 
#> CI Method:         percentile 
#> Replicates (B):    100 
#> 
#>    index estimate     method          type alpha conf_level  lower  upper
#> 1    Cpy   0.9972 percentile nonparametric  0.10        90% 0.9807 1.0019
#> 2    Cpy   0.9972 percentile nonparametric  0.05        95% 0.9776 1.0024
#> 3    Cpy   0.9972 percentile nonparametric  0.01        99% 0.9738 1.0026
#> 4     Cp   0.9262 percentile nonparametric  0.10        90% 0.7712 1.1300
#> 5     Cp   0.9262 percentile nonparametric  0.05        95% 0.7664 1.2478
#> 6     Cp   0.9262 percentile nonparametric  0.01        99% 0.7424 1.2855
#> 7    Cpk   0.9098 percentile nonparametric  0.10        90% 0.6999 1.0734
#> 8    Cpk   0.9098 percentile nonparametric  0.05        95% 0.6859 1.1631
#> 9    Cpk   0.9098 percentile nonparametric  0.01        99% 0.6528 1.2616
#> 10   Cpm   0.9251 percentile nonparametric  0.10        90% 0.7636 1.1135
#> 11   Cpm   0.9251 percentile nonparametric  0.05        95% 0.7470 1.2032
#> 12   Cpm   0.9251 percentile nonparametric  0.01        99% 0.7280 1.2822
#> 13  Cpmk   0.9087 percentile nonparametric  0.10        90% 0.6665 1.0714
#> 14  Cpmk   0.9087 percentile nonparametric  0.05        95% 0.6536 1.1306
#> 15  Cpmk   0.9087 percentile nonparametric  0.01        99% 0.6120 1.2583
#> 16  Spmk   0.9240 percentile nonparametric  0.10        90% 0.7305 1.0930
#> 17  Spmk   0.9240 percentile nonparametric  0.05        95% 0.7164 1.1633
#> 18  Spmk   0.9240 percentile nonparametric  0.01        99% 0.6842 1.2789
#> 19  CpTk   0.9607 percentile nonparametric  0.10        90% 0.6724 0.9728
#> 20  CpTk   0.9607 percentile nonparametric  0.05        95% 0.6397 0.9873
#> 21  CpTk   0.9607 percentile nonparametric  0.01        99% 0.6060 0.9991
#> 22 CNpmc   0.7601 percentile nonparametric  0.10        90% 0.6627 0.8547
#> 23 CNpmc   0.7601 percentile nonparametric  0.05        95% 0.6516 0.8932
#> 24 CNpmc   0.7601 percentile nonparametric  0.01        99% 0.6389 0.9242
#>     width
#> 1  0.0212
#> 2  0.0248
#> 3  0.0288
#> 4  0.3589
#> 5  0.4814
#> 6  0.5431
#> 7  0.3736
#> 8  0.4772
#> 9  0.6088
#> 10 0.3499
#> 11 0.4562
#> 12 0.5542
#> 13 0.4049
#> 14 0.4770
#> 15 0.6464
#> 16 0.3625
#> 17 0.4470
#> 18 0.5947
#> 19 0.3004
#> 20 0.3476
#> 21 0.3931
#> 22 0.1920
#> 23 0.2416
#> 24 0.2853

5. Diagnostics: SE, MSE, and Coverage Probabilities

diag_res <- compute_diagnostics_censor(
  fit = fit_cap,
  true_params = list(mean = 10, sd = 1.5),
  true_indices = c(Cpy = 1.0, Cp = 1.33),
  B = 50
)
#> Warning in stats::pnorm(x, mean, sd): NaNs produced
#> Warning in stats::pnorm(x, mean, sd): NaNs produced
#> Warning in stats::pnorm(x, mean, sd): NaNs produced
#> Warning in stats::pnorm(x, mean, sd): NaNs produced
#> Warning in norm.inter(t, alpha): extreme order statistics used as endpoints
#> Warning in norm.inter(t, alpha): extreme order statistics used as endpoints
#> Warning in norm.inter(t, alpha): extreme order statistics used as endpoints
#> Warning in norm.inter(t, alpha): extreme order statistics used as endpoints
#> Warning in norm.inter(t, alpha): extreme order statistics used as endpoints
#> Warning in norm.inter(t, alpha): extreme order statistics used as endpoints
#> Warning in norm.inter(t, alpha): extreme order statistics used as endpoints
#> Warning in norm.inter(t, alpha): extreme order statistics used as endpoints

print(diag_res)
#> --- Diagnostics for Interval-Censored Data ---
#> Bootstrap Replicates: 50 
#> 
#> Standard Errors for Parameters:
#>     mean       sd 
#> 0.049890 0.030722 
#> 
#> Mean Squared Errors for Parameters:
#>     mean       sd 
#> 0.004992 0.003651 
#> 
#> Standard Errors for Capability Indices:
#>      Cpy       Cp      Cpk      Cpm     Cpmk     Spmk     CpTk    CNpmc 
#> 0.006762 0.151569 0.151158 0.148054 0.152630 0.148090 0.092541 0.074686 
#> 
#> Mean Squared Errors for Capability Indices:
#>  Cpy.Cpy    Cp.Cp      Cpk      Cpm     Cpmk     Spmk     CpTk    CNpmc 
#> 0.000008 0.163056 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 
#> 
#> Coverage Probabilities for Capability Indices:
#> 
#> Alpha level: alpha_0.1 
#>   Cpy    Cp   Cpk   Cpm  Cpmk  Spmk  CpTk CNpmc 
#>     1     0     0     0     0     0     0     0 
#> 
#> Alpha level: alpha_0.05 
#>   Cpy    Cp   Cpk   Cpm  Cpmk  Spmk  CpTk CNpmc 
#>     1     0     0     0     0     0     0     0 
#> 
#> Alpha level: alpha_0.01 
#>   Cpy    Cp   Cpk   Cpm  Cpmk  Spmk  CpTk CNpmc 
#>     1     1     0     0     0     0     0     0 
#> 
#> Coverage Probabilities for Parameters:
#> 
#> Alpha level: alpha_0.1 
#> mean   sd 
#>    1    0 
#> 
#> Alpha level: alpha_0.05 
#> mean   sd 
#>    1    0 
#> 
#> Alpha level: alpha_0.01 
#> mean   sd 
#>    1    1

6. Visualization

plot(fit_cap)

References

Maiti, S. S., Saha, M., & Nanda, A. K. (2010). On Generalizing Process Capability Indices. Quality Technology & Quantitative Management, 7(3), 279-300.
Saha, M., Dey, S., & Maiti, S. S. (2018). Parametric and non-parametric bootstrap confidence intervals of CNpk for exponential power distribution. Journal of Industrial and Production Engineering, 35(3), 160-169.
Dey, S., & Saha, M. (2019). Assessing the process capability index Spmk using improved estimators. Life Cycle Reliability and Safety Engineering, 8(3), 253-264.
Saha, M., Dey, S., & Maiti, S. S. (2019). Bootstrap confidence intervals of CpTk for two parameter logistic exponential distribution with applications. International Journal of System Assurance Engineering and Management, 10(4), 861-872.
Alotaibi, R., Dey, S., & Saha, M. (2022). Estimation and Confidence Intervals of a New PCI CNpmc for Logistic-Exponential Process Distribution. Journal of Mathematics, 2022, 3135264.
Saha, M., Dey, S., & Nadarajah, S. (2022). Parametric inference of the process capability index Cpc for exponentiated exponential distribution. Journal of Applied Statistics, 49(16), 4097-4121.
Saha, M., Tripathi, H., & Dey, S. (2024). Classical Inference of a New PCI CNpmkc for Logistic-Exponential Process Distribution. International Journal of Reliability, Quality and Safety Engineering, 31(3), 2450013.

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