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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 \(T^*\).
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 TRUEThe 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 0.3686470Goodness-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: 0.70588These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.
Health stats visible at Monitor.