Author: Shikhar Tyagi
Date: 2026-08-16

# Bayesian Estimation of Generalized Process Capability Indices for Hybrid Type-II Censored Data via Importance Sampling

#### Shikhar Tyagi

#### 2026-08-16

# Introduction

The `gpcihybridIIImpSam` package provides a comprehensive, generalized Bayesian framework for evaluating Generalized Process Capability Indices (GPCIs) under Hybrid Type-II Censored Lifetime Data using Importance Sampling (Sampling Importance Resampling, SIR).

Under Hybrid Type-II censoring, \(n\) identical units are placed on life test. The experiment terminates at: \[T^* = \max(x_r, T_c)\] where \(r \le n\) is the target number of failures and \(T_c > 0\) is the pre-fixed censoring time. The likelihood function is: \[L(\theta \mid \mathbf{x}, r, T_c, n) = \left[ \prod_{i=1}^d f(x_i; \theta) \right] [S(T^*; \theta)]^{n - d}\] where \(d \ge r\) is the number of observed failures up to time \(T^*\).

# Getting Started

    library(gpcihybridIIImpSam)

    # 1. User-supplied custom probability density, CDF, and survival functions
    my_pdf  <- function(x, rate) stats::dexp(x, rate = rate)
    my_cdf  <- function(q, rate) stats::pexp(q, rate = rate)
    my_surv <- function(q, rate) stats::pexp(q, rate = rate, lower.tail = FALSE)

    # 2. Observed Hybrid Type-II Censored Data
    # n = 10 units placed on test, target r = 3, fixed censoring time Tc = 2.5
    x_data <- c(0.4, 0.9, 1.5, 2.3)

    # 3. Fit Importance Sampling Model
    fit <- gpci_hybrid2_impsam(
      x = x_data,
      r = 3,
      tc = 2.5,
      n = 10,
      pdf = my_pdf,
      cdf = my_cdf,
      surv = my_surv,
      param_names = "rate",
      start = c(rate = 0.5),
      chain_length = 500,
      burn_in = 100,
      thinning = 1,
      USL = 8,
      LSL = 0,
      target = 4,
      indices = c("Cpy", "Cp", "Cpk", "Cpm", "CNpmc")
    )

    # 4. Print Summary
    print(fit)
    #> --- Bayesian Importance Sampling of GPCIs under Hybrid Type-II Censoring ---
    #> Distribution:  Custom
    #> Observed Failures (d):  4  | Target (r):  3  | Total (n):  10
    #> Censoring Time (Tc):  2.5  | Effective Termination (T*):  2.5
    #> Specification Limits: LSL = 0 | Target = 4 | USL = 8
    #>
    #> Initial Parameter MLEs (under Hybrid Type-II Censoring):
    #>  rate
    #> 0.199
    #>
    #> Point Estimates of GPCIs (at Initial Estimates):
    #>    Cpy     Cp    Cpk    Cpm  CNpmc
    #> 0.7987 0.2654 0.1974 0.2600 0.2362
    #>
    #> Posterior GPCI Summary (Importance Sampling Chain):
    #>   Variable Estimate    Bias    MSE Bayes_Risk HPD_95_Lower HPD_95_Upper
    #> 1      Cpy   0.7267 -0.0720 0.0347     0.0296       0.4235       0.9743
    #> 2       Cp   0.2553 -0.0101 0.0165     0.0164       0.0540       0.5376
    #> 3      Cpk   0.1268 -0.0706 0.0350     0.0301      -0.1504       0.3333
    #> 4      Cpm   0.2382 -0.0218 0.0145     0.0141       0.0740       0.4679
    #> 5    CNpmc   0.2113 -0.0249 0.0084     0.0078       0.0775       0.3686
    #>   HW_Passed
    #> 1      TRUE
    #> 2      TRUE
    #> 3      TRUE
    #> 4      TRUE
    #> 5      TRUE

# Statistical Summary and Diagnostics

The package automatically reports: - Posterior point estimate (posterior mean) - Bias and Mean Squared Error (MSE) - Bayes risk under squared error loss (posterior variance) - Highest Posterior Density (HPD) credible intervals at 90%, 95%, and 99% levels - Heidelberger and Welch’s MCMC convergence diagnostics (stationarity test and half-width test) - Convergence probability and empirical coverage probability

    # Full posterior summary
    summary(fit)
    #> ======================================================================
    #>  Bayesian Importance Sampling of GPCIs under Hybrid Type-II Censoring
    #> ======================================================================
    #>
    #> --- Parameter Posterior Summary ---
    #>   Variable  Estimate         Bias         MSE  Bayes_Risk HPD_90_Lower
    #> 1     rate 0.1914652 -0.007558266 0.009263195 0.009224517   0.04861269
    #>   HPD_90_Upper HPD_95_Lower HPD_95_Upper HPD_99_Lower HPD_99_Upper    HW_Stat
    #> 1    0.3217912   0.04050443    0.4032326   0.03893146    0.4455496 0.08688201
    #>   HW_PValue HW_Passed Convergence_Prob Coverage_Prob
    #> 1       0.5      TRUE              0.5             1
    #>
    #> --- GPCI Posterior Summary ---
    #>   Variable  Estimate        Bias        MSE  Bayes_Risk HPD_90_Lower
    #> 1      Cpy 0.7267080 -0.07196828 0.03467797 0.029557649   0.48452096
    #> 2       Cp 0.2552869 -0.01007769 0.01646790 0.016399141   0.06481692
    #> 3      Cpk 0.1267932 -0.07060260 0.03504495 0.030120468  -0.11329193
    #> 4      Cpm 0.2381709 -0.02184335 0.01450207 0.014053046   0.05047659
    #> 5    CNpmc 0.2112692 -0.02490033 0.00835838 0.007753861   0.09465742
    #>   HPD_90_Upper HPD_95_Lower HPD_95_Upper HPD_99_Lower HPD_99_Upper    HW_Stat
    #> 1    0.9743164   0.42352318    0.9743164   0.26834146    0.9743164 0.15867759
    #> 2    0.4290549   0.05400590    0.5376435   0.05190861    0.5940662 0.08688198
    #> 3    0.3333333  -0.15038712    0.3333333  -0.22951611    0.3333333 0.15237804
    #> 4    0.4123882   0.07403715    0.4679232   0.03966306    0.4679232 0.10042807
    #> 5    0.3686470   0.07752096    0.3686470   0.04234584    0.3686470 0.11818107
    #>   HW_PValue HW_Passed Convergence_Prob Coverage_Prob
    #> 1       0.5      TRUE              0.5             1
    #> 2       0.5      TRUE              0.5             1
    #> 3       0.5      TRUE              0.5             1
    #> 4       0.5      TRUE              0.5             1
    #> 5       0.5      TRUE              0.5             1
    #>
    #> --- Effective Sample Ratio / Acceptance Rates ---
    #>   rate
    #> 0.3067

    # Extract 95% HPD credible intervals for capability indices
    confint(fit, what = "indices", level = 0.95)
    #>              2.5%     97.5%
    #> Cpy    0.42352318 0.9743164
    #> Cp     0.05400590 0.5376435
    #> Cpk   -0.15038712 0.3333333
    #> Cpm    0.07403715 0.4679232
    #> CNpmc  0.07752096 0.3686470

# Goodness-of-Fit Testing

Goodness-of-fit testing for Hybrid Type-II censored data is supported via `gofPHCS`:

    dist_exp <- dist_exponential(rate = 0.5)
    gof_res <- gof_test_hybrid2(
      fit = fit,
      statistic = "auto",
      p.method = "montecarlo",
      nsim = 50
    )
    print(gof_res)
    #> --- Goodness-of-Fit Test (Hybrid Type-II Censored Data) ---
    #> Distribution:  Custom
    #> Test Method:   Monte Carlo Test (50 replicates) (KSII for hybridII scheme)
    #> Statistic:     KSII = 1.0361
    #> p-value:       0.70588

# References

1. Alotaibi, N., Elbatal, I., Shrahili, M., et al. (2022). Estimation and Confidence Intervals of a New PCI \(C_{Npmc}\) for Logistic-Exponential. Journal of Mathematics, 2022:3135264. doi:10.1155/2022/3135264.
2. Childs, A., Chandrasekar, B., Balakrishnan, N., & Kundu, D. (2003). Exact likelihood inference based on Type-I and Type-II hybrid censored samples from the exponential distribution. Annals of the Institute of Statistical Mathematics, 55(2), 319-330. doi:10.1080/0266476032000053637.
3. Dey, S., & Saha, M. (2019). The generalized process capability index \(S_{pmk}\) for Gompertz distribution. Quality and Reliability Engineering International, 35(8), 2603-2615. doi:10.1007/s41872-019-00081-4.
4. Kundu, D., & Pradhan, B. (2009). Estimating the parameters of the generalized exponential distribution in presence of hybrid censoring. Communications in Statistics - Simulation and Computation, 38(10), 2030-2041. doi:10.1016/j.spl.2008.09.006.
5. Maiti, S. S., Saha, M., & Nanda, A. K. (2010). On generalizing process capability indices. Quality Technology & Quantitative Management, 7(3), 279-300. doi:10.1080/16843703.2010.11673233.
6. Saha, M., Smarandache, F., & Dey, S. (2022). Process capability index \(C_{pc}\) for lifetime data. Journal of Applied Statistics, 49(8), 2011-2032. doi:10.1080/02664763.2021.1971632.
7. Saha, M., Tyagi, S., et al. (2024). Generalized process capability index \(C_{Npmkc}\) for quality evaluation. International Journal of Quality & Reliability Management. doi:10.1142/S021853932450013X.