Lindley Approximation for Capability Indices under Hybrid Censoring

library(gpcihybridIILinApp)

Introduction

The gpcihybridIILinApp package provides a comprehensive framework for computing, estimating, and validating Generalized Process Capability Indices (GPCIs) using the Lindley Approximation Method for Hybrid Type-II censored lifetime data under Bayesian inference.

Hybrid Type-II Censoring Scheme

Under Hybrid Type-II censoring with initial sample size \(n\), target failures \(r\), and censoring time \(T_c\), the experiment terminates at time: \[T^* = \max(x_r, T_c)\] The likelihood function based on observed failure times \(x = (x_1, \dots, x_d)\) (\(d \ge r\)) is: \[L(\theta \mid x, r, T_c, n) = \left[ \prod_{i=1}^d f(x_i; \theta) \right] [S(T^*; \theta)]^{n-d}\]

Supported Generalized Process Capability Indices

Supported GPCIs include: - \(C_{py}\) (Process Capability Index based on Yield; Maiti et al., 2010) - \(C_p, C_{pk}, C_{pu}, C_{pl}, C_{pm}, C_{pmk}\) (Classical capability indices) - \(C_{pTk}\) (Saha et al., 2019) - \(S_{pmk}\) (Dey & Saha, 2019) - \(C_{pc}\) (Saha et al., 2022) - \(CN_{pk}\) (Saha et al., 2018) - \(CN_{pmc}\) (Alotaibi et al., 2022) - \(CN_{pmkc}\) (Saha et al., 2024) - \(C_p(u,v), CN_p(u,v)\) (Vännman’s generalized family) - Quantile-based analogs (\(C_{p,q}, C_{pk,q}, C_{pu,q}, C_{pl,q}, C_{pm,q}, C_{pmk,q}\))

Basic Example with Custom PDF/CDF/SF

Below is an example estimating GPCIs for Hybrid Type-II censored data using the high-level user interface function gpci_lindley_hybrid2():

set.seed(42)
x_obs <- c(0.2, 0.5, 0.8, 1.1, 1.4)

fit_res <- gpci_lindley_hybrid2(
  x = x_obs, r = 3, tc = 1.2, n = 10,
  pdf = function(x, rate = 1) stats::dexp(x, rate = rate),
  cdf = function(x, rate = 1) stats::pexp(x, rate = rate),
  sf  = function(x, rate = 1) stats::pexp(x, rate = rate, lower.tail = FALSE),
  chain_length = 50,
  burn_in = 10,
  thinning = 1,
  USL = 3,
  LSL = 0,
  target = 1.5,
  indices = c("Cpy", "Cp", "Cpk", "Cpm", "CNpmc"),
  B = 20
)

# Display Summary Diagnostics Table
summary(fit_res)
#>   Index MLE_Estimate Lindley_Estimate   Chain_Mean Bias MSE Risk_Value
#> 1   Cpy 9.826530e-01     9.826530e-01 9.826530e-01    0   0          0
#> 2    Cp 5.000000e+03     5.000000e+03 5.000000e+03    0   0          0
#> 3   Cpk 0.000000e+00     0.000000e+00 0.000000e+00    0   0          0
#> 4   Cpm 3.333333e-01     3.333333e-01 3.333333e-01    0   0          0
#> 5 CNpmc 2.328306e-11     2.328306e-11 2.328306e-11    0   0          0
#>    HPD90_Lower  HPD90_Upper  HPD95_Lower  HPD95_Upper  HPD99_Lower  HPD99_Upper
#> 1 9.826530e-01 9.826530e-01 9.826530e-01 9.826530e-01 9.826530e-01 9.826530e-01
#> 2 5.000000e+03 5.000000e+03 5.000000e+03 5.000000e+03 5.000000e+03 5.000000e+03
#> 3 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> 4 3.333333e-01 3.333333e-01 3.333333e-01 3.333333e-01 3.333333e-01 3.333333e-01
#> 5 2.328306e-11 2.328306e-11 2.328306e-11 2.328306e-11 2.328306e-11 2.328306e-11
#>   HW_Stat HW_Pvalue HW_Passed Convergence_Prob Boot95_Lower Boot95_Upper
#> 1       0       0.5      TRUE              0.5    0.9527853    0.9527853
#> 2       0       0.5      TRUE              0.5    0.5123905    0.5123905
#> 3       0       0.5      TRUE              0.5    0.3388922    0.3388922
#> 4       0       0.5      TRUE              0.5    0.4545094    0.4545094
#> 5       0       0.5      TRUE              0.5    0.2954846    0.2954846

Using Built-in Distribution Objects

The package provides pre-defined distribution objects such as dist_exponential(), dist_weibull(), dist_gamma(), dist_normal(), dist_lognormal(), dist_logistic(), and dist_loglogistic():

dist_exp <- dist_exponential(rate = 1)

fit_exp <- lindley_gpci_hybrid2(
  x = x_obs, r = 3, tc = 1.2, n = 10,
  distribution = dist_exp,
  chain_length = 50,
  burn_in = 10,
  thinning = 1,
  USL = 3,
  LSL = 0,
  target = 1.5,
  indices = c("Cpy", "Cp", "Cpk", "Cpm", "CNpmc"),
  B = 20
)

summary(fit_exp)
#>   Index MLE_Estimate Lindley_Estimate Chain_Mean       Bias         MSE
#> 1   Cpy    0.7789735        0.7960890  0.7926165 0.01364305 0.019114971
#> 2    Cp    0.2500004        0.3002505  0.2956360 0.04563563 0.014370784
#> 3   Cpk    0.1666675        0.2671676  0.2030214 0.03635391 0.024855649
#> 4   Cpm    0.2425361        0.2978745  0.2830799 0.04054378 0.012987002
#> 5 CNpmc    0.2065101        0.2225041  0.2188001 0.01229003 0.003819121
#>    Risk_Value HPD90_Lower HPD90_Upper HPD95_Lower HPD95_Upper HPD99_Lower
#> 1 0.019301528  0.59712146   0.9611374   0.5203717   0.9611374  0.36923108
#> 2 0.016469326  0.07653943   0.4312088   0.1219698   0.5305144  0.07653943
#> 3 0.026185300 -0.03162445   0.3333333  -0.0893937   0.3333333 -0.18025447
#> 4 0.014641965  0.13232403   0.4566070   0.1030072   0.4566070  0.06063325
#> 5 0.003970476  0.12929864   0.2991386   0.1038689   0.2991386  0.06354932
#>   HPD99_Upper    HW_Stat HW_Pvalue HW_Passed Convergence_Prob Boot95_Lower
#> 1   0.9611374 0.04717302       0.5      TRUE              0.5   0.58973735
#> 2   0.5305144 0.05605243       0.5      TRUE              0.5   0.14800487
#> 3   0.3333333 0.05114820       0.5      TRUE              0.5  -0.03732359
#> 4   0.4566070 0.05641950       0.5      TRUE              0.5   0.12940756
#> 5   0.2991386 0.05005141       0.5      TRUE              0.5   0.12679321
#>   Boot95_Upper
#> 1    0.9400568
#> 2    0.4627408
#> 3    0.3333333
#> 4    0.4309856
#> 5    0.2895360

Goodness-of-Fit Testing

Goodness-of-fit testing under Hybrid Type-II censoring can be performed directly using gof_test_hybrid2():

gof_res <- gof_test_hybrid2(
  fit = fit_exp,
  p.method = "asymptotic"
)

print(gof_res)
#> --- Goodness-of-Fit Test (Hybrid Type-II Censored Data) ---
#> Distribution:  Exponential 
#> Test Method:   Error in gofPHCS::gof_test: Asymptotic p-value method is not available for statistic 'KSII'. Use p.method = 'montecarlo'. 
#> Statistic:     Error = 0 
#> p-value:       1

References

  1. Lindley, D. V. (1980). Approximate Bayesian methods. Trabajos de Estadística y de Investigación Operativa, 31(1), 223-245.
  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.
  3. Balakrishnan, N., & Kundu, D. (2013). Hybrid censoring: models, methods and applications. Naval Research Logistics, 60(5), 379-409.
  4. Maiti, S. S., Saha, M., & Nanda, A. K. (2010). On generalizing process capability indices. Quality Technology & Quantitative Management, 7(3), 279-289.
  5. 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.
  6. Dey, S., & Saha, M. (2019). Assessing the process capability index Spmk using improved estimators. Life Cycle Reliability and Safety Engineering, 8, 81-88.
  7. 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.
  8. 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.
  9. 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.
  10. Saha, M., Tripathi, V., & 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.
  11. Heidelberger, P., & Welch, P. D. (1983). Simulation run length control in the presence of an initial transient. Operations Research, 31(6), 1109-1144.