Package {gpcihybridIILinApp}

Type:	Package
Title:	Lindley Approximation for Capability Indices under Hybrid Censoring
Version:	0.1.0
Description:	Provides a comprehensive framework for estimating Generalized Process Capability Indices (GPCIs) under Hybrid Type-II censored lifetime data using Lindley's 3rd-order approximation method (Lindley, 1980 <doi:10.2307/2345271>). Supports user-supplied probability density/mass functions (PDF/PMF), cumulative distribution functions (CDF), survival functions (SF), and quantile functions. Computes Maximum Likelihood Estimates (MLE) using the 'MleCensoR' package (Childs et al., 2003 <doi:10.1007/BF02517803>; Balakrishnan & Kundu, 2013 <doi:10.1002/nav.21545>) and Bayesian posterior expectations for classical and non-normal capability indices, including Cpy (Maiti et al., 2010 <doi:10.1080/16843703.2010.11673233>), Spmk (Dey & Saha, 2019 <doi:10.1007/s41872-019-00081-4>), CpTk (Saha et al., 2019), Cpc (Saha et al., 2022 <doi:10.1080/02664763.2021.1971632>), CNpmc (Alotaibi et al., 2022 <doi:10.1155/2022/3135264>), CNpmkc (Saha et al., 2024 <doi:10.1142/S021853932450013X>), CNpk (Saha et al., 2018 <doi:10.1080/21681015.2018.1437793>), and Vannman's Cp(u,v) family. Generates posterior parameter and GPCI chains via sampling with burn-in and thinning, calculating Bias, Mean Squared Error (MSE), Bayes Risk (SEL and Linex), Highest Posterior Density (HPD) credible intervals at 90%, 95%, and 99% levels, and Heidelberger and Welch's MCMC Convergence Diagnostics (Heidelberger & Welch, 1983 <doi:10.1287/opre.31.6.1109>) with convergence probabilities. Evaluates parametric and non-parametric bootstrap confidence intervals (Percentile, Normal, Basic, BCp, BCa) at 90%, 95%, and 99% levels of significance. Integrates goodness-of-fit testing for Hybrid Type-II censored data via the 'gofPHCS' package.
License:	MIT + file LICENSE
Encoding:	UTF-8
RoxygenNote:	7.3.3
Depends:	R (≥ 4.0.0)
Imports:	stats, graphics, numDeriv, MleCensoR, gofPHCS
Suggests:	testthat (≥ 3.0.0), knitr, rmarkdown
VignetteBuilder:	knitr
Config/testthat/edition:	3
NeedsCompilation:	no
Packaged:	2026-08-16 23:25:40 UTC; shikhar tyagi
Author:	Shikhar Tyagi [aut, cre], Sumit Kumar [aut], Arvind Pandey [aut], Bhupendra Singh [aut], Vrijesh Tripathi [aut]
Maintainer:	Shikhar Tyagi <shikhar1093tyagi@gmail.com>
Repository:	CRAN
Date/Publication:	2026-08-21 13:00:20 UTC

gpcihybridIILinApp: Lindley Approximation for Capability Indices under Hybrid Censoring

Description

A comprehensive package for estimating and analyzing Generalized Process Capability Indices (GPCIs) under Hybrid Type-II censored lifetime data using Lindley's 3rd-order approximation method. Provides Maximum Likelihood Estimation (MLE) with 'MleCensoR' integration, Bayesian Lindley approximation point estimates, MCMC-style posterior chain generation with burn-in and thinning, statistical diagnostic metrics (Bias, MSE, Bayes Risk, HPD intervals, Heidelberger-Welch convergence diagnostics, convergence/coverage probabilities), parametric and non-parametric bootstrap confidence intervals (Percentile, Normal, Basic, BCp, BCa), and goodness-of-fit testing with 'gofPHCS'.

Author(s)

Maintainer: Shikhar Tyagi shikhar1093tyagi@gmail.com (ORCID)

Authors:

• Sumit Kumar stats.sumitbhal@gmail.com

• Arvind Pandey arvindmzu@gmail.com

• Bhupendra Singh bhupendra.rana@gmail.com

• Vrijesh Tripathi vrijesh.tripathi@uwi.edu

Parametric and Non-Parametric Bootstrap Confidence Intervals under Hybrid Type-II Censoring

Description

Computes bootstrap confidence intervals for Generalized Process Capability Indices (GPCIs) under Hybrid Type-II censored lifetime data using parametric or non-parametric resampling. Supports Percentile, Normal, Basic, and BCp (Bias-Corrected Percentile) bootstrap methods at 90

Usage

bootstrap_hybrid2_gpci(
  x,
  r,
  tc,
  n,
  distribution = NULL,
  USL,
  LSL,
  target = (USL + LSL)/2,
  indices = NULL,
  type = c("parametric", "nonparametric"),
  B = 500,
  conf_levels = c(0.9, 0.95, 0.99),
  u = 1,
  v = 1,
  ...
)

boot_ci_hybrid2(
  x,
  r,
  tc,
  n,
  distribution = NULL,
  USL,
  LSL,
  target = (USL + LSL)/2,
  indices = NULL,
  type = c("parametric", "nonparametric"),
  B = 500,
  conf_levels = c(0.9, 0.95, 0.99),
  u = 1,
  v = 1,
  ...
)

bootstrap_gpci_hybrid2(
  x,
  r,
  tc,
  n,
  distribution = NULL,
  USL,
  LSL,
  target = (USL + LSL)/2,
  indices = NULL,
  type = c("parametric", "nonparametric"),
  B = 500,
  conf_levels = c(0.9, 0.95, 0.99),
  u = 1,
  v = 1,
  ...
)

Arguments

Details

Under Hybrid Type-II censoring, bootstrap samples of initial size n are generated. For parametric bootstrap, n observations are simulated from the fitted distribution, sorted in ascending order y_1 \le y_2 \le \dots \le y_n, and the experiment termination time T^* = \max(y_r, T_c) is applied to select observed failures y_i \le T^*.

Value

An object of S3 class "gpciBootstrapHybrid2" containing a list with:

Examples

dist_exp <- dist_exponential(rate = 1)
x_data <- c(0.2, 0.5, 0.8, 1.1)
boot_res <- bootstrap_hybrid2_gpci(
  x = x_data, r = 3, tc = 1.0, n = 10,
  distribution = dist_exp, USL = 3, LSL = 0,
  indices = c("Cpy", "Cp"), B = 5
)
print(boot_res)

Compute Process Capability Indices under Hybrid Type-II Censored Data

Description

Computes classical and generalized Process Capability Indices (GPCIs) under Hybrid Type-II censored lifetime data.

Usage

capability_hybrid2(
  x = NULL,
  r = NULL,
  tc = NULL,
  n = NULL,
  distribution,
  USL,
  LSL,
  target = (USL + LSL)/2,
  indices = c("Cpy", "Spmk", "CpTk", "Cpc", "CNpmc", "CNpmkc", "CNpk", "Cp", "Cpk",
    "Cpm", "Cpmk"),
  u = 1,
  v = 1,
  mode = c("moments", "quantile"),
  fit = TRUE,
  start = NULL,
  C0 = 1,
  C1 = 0,
  C2 = 1,
  tolerance_t = USL - LSL,
  P0 = 0.9973002,
  LDL = LSL,
  UDL = USL,
  ...
)

gpci_fit_hybrid2(
  x = NULL,
  r = NULL,
  tc = NULL,
  n = NULL,
  distribution,
  USL,
  LSL,
  target = (USL + LSL)/2,
  indices = c("Cpy", "Spmk", "CpTk", "Cpc", "CNpmc", "CNpmkc", "CNpk", "Cp", "Cpk",
    "Cpm", "Cpmk"),
  u = 1,
  v = 1,
  mode = c("moments", "quantile"),
  fit = TRUE,
  start = NULL,
  C0 = 1,
  C1 = 0,
  C2 = 1,
  tolerance_t = USL - LSL,
  P0 = 0.9973002,
  LDL = LSL,
  UDL = USL,
  ...
)

capability(
  x = NULL,
  r = NULL,
  tc = NULL,
  n = NULL,
  distribution,
  USL,
  LSL,
  target = (USL + LSL)/2,
  indices = c("Cpy", "Spmk", "CpTk", "Cpc", "CNpmc", "CNpmkc", "CNpk", "Cp", "Cpk",
    "Cpm", "Cpmk"),
  u = 1,
  v = 1,
  mode = c("moments", "quantile"),
  fit = TRUE,
  start = NULL,
  C0 = 1,
  C1 = 0,
  C2 = 1,
  tolerance_t = USL - LSL,
  P0 = 0.9973002,
  LDL = LSL,
  UDL = USL,
  ...
)

Arguments

Details

Evaluates classical and generalized process capability indices for arbitrary continuous process distributions under Hybrid Type-II censoring. Supported indices include:

• C_{py}: Yield-based capability index (Maiti et al., 2010).

• S_{pmk}: Generalized loss-based index (Dey & Saha, 2019).

• C_{pTk}: Asymmetric target-based index (Saha et al., 2019).

• C_{pc}: Loss-based capability index (Saha et al., 2022).

• C_{Npmc}: Tolerance-loss based index (Alotaibi et al., 2022).

• C_{Npmkc}: Generalized asymmetric index (Saha et al., 2024).

• C_{Npk}: Non-normal capability index (Saha et al., 2018).

• C_p, C_{pk}, C_{pu}, C_{pl}, C_{pm}, C_{pmk}: Classical process capability indices.

• C_p(u,v), C_{Np}(u,v): Vännman's generalized family of capability indices.

• Quantile-based capability indices (C_{p,q}, C_{pk,q}, C_{pu,q}, C_{pl,q}, C_{pm,q}, C_{pmk,q}).

Value

An S3 object of class "gpcifit_hybrid2" containing:

Examples

dist_exp <- dist_exponential(rate = 1)
x <- c(0.2, 0.5, 0.8, 1.1)
res <- capability_hybrid2(
  x = x, r = 3, tc = 1.0, n = 10,
  distribution = dist_exp,
  USL = 3, LSL = 0, target = 1.5,
  indices = c("Cpy", "Spmk", "CpTk", "Cpc", "CNpmc", "CNpmkc", "Cp", "Cpk", "Cpm", "Cpmk")
)
print(res)

Coef Method for gpcifit_hybrid2

Description

Extract point estimates of capability indices or fitted distribution parameters.

Usage

## S3 method for class 'gpcifit_hybrid2'
coef(object, what = c("indices", "parameters"), ...)

Arguments

Value

Named numeric vector of indices or parameters.

Compute Theoretical Moments for a Process Distribution

Description

Compute Theoretical Moments for a Process Distribution

Usage

compute_theoretical_moments_hybrid2(distribution)

Arguments

Value

A list containing mean, var, skewness, and kurtosis.

Examples

dist_norm <- dist_normal(mean = 10, sd = 2)
compute_theoretical_moments_hybrid2(dist_norm)

Define a Process Distribution for Hybrid Type-II Censored Data

Description

Constructor to define a continuous or discrete probability distribution for process capability analysis under Hybrid Type-II censored data. The distribution can be defined by specifying any one (or more) of its Probability Density/Mass Function (PDF/PMF), Cumulative Distribution Function (CDF), Survival Function (SF), or Quantile Function. Missing functions are automatically and numerically derived using adaptive integration and root-finding algorithms.

Usage

define_distribution_hybrid2(
  name,
  pdf = NULL,
  cdf = NULL,
  sf = NULL,
  quantile = NULL,
  params = list(),
  support = c(-Inf, Inf)
)

gpci_dist_hybrid2(
  name,
  pdf = NULL,
  cdf = NULL,
  sf = NULL,
  quantile = NULL,
  params = list(),
  support = c(-Inf, Inf)
)

define_distribution(
  name,
  pdf = NULL,
  cdf = NULL,
  sf = NULL,
  quantile = NULL,
  params = list(),
  support = c(-Inf, Inf)
)

Arguments

Value

An S3 object of class "gpci_dist_hybrid2" containing:

Examples

# Define custom Weibull distribution using only the survival function
custom_weib <- define_distribution_hybrid2(
  name = "custom_weibull",
  sf = function(x, shape, scale) stats::pweibull(x, shape, scale, lower.tail = FALSE),
  params = list(shape = 2, scale = 10),
  support = c(0, Inf)
)
print(custom_weib)

Exponential Distribution Constructor

Description

Exponential Distribution Constructor

Usage

dist_exponential(rate = 1)

Arguments

Value

A gpci_dist_hybrid2 object for Exponential distribution.

Examples

dist_e <- dist_exponential(rate = 0.5)

Gamma Distribution Constructor

Description

Gamma Distribution Constructor

Usage

dist_gamma(shape = 1, scale = 1)

Arguments

Value

A gpci_dist_hybrid2 object for Gamma distribution.

Examples

dist_g <- dist_gamma(shape = 3, scale = 2)

Logistic Distribution Constructor

Description

Logistic Distribution Constructor

Usage

dist_logistic(location = 0, scale = 1)

Arguments

Value

A gpci_dist_hybrid2 object for Logistic distribution.

Examples

dist_l <- dist_logistic(location = 0, scale = 1)

Log-Logistic Distribution Constructor

Description

Log-Logistic Distribution Constructor

Usage

dist_loglogistic(shape = 1, scale = 1)

Arguments

Value

A gpci_dist_hybrid2 object for Log-Logistic distribution.

Examples

dist_ll <- dist_loglogistic(shape = 2, scale = 3)

Lognormal Distribution Constructor

Description

Lognormal Distribution Constructor

Usage

dist_lognormal(meanlog = 0, sdlog = 1)

Arguments

Value

A gpci_dist_hybrid2 object for Lognormal distribution.

Examples

dist_ln <- dist_lognormal(meanlog = 1, sdlog = 0.5)

Normal Distribution Constructor

Description

Normal Distribution Constructor

Usage

dist_normal(mean = 0, sd = 1)

Arguments

Value

A gpci_dist_hybrid2 object for Normal distribution.

Examples

dist_norm <- dist_normal(mean = 10, sd = 2)

Weibull Distribution Constructor

Description

Weibull Distribution Constructor

Usage

dist_weibull(shape = 1, scale = 1)

Arguments

Value

A gpci_dist_hybrid2 object for Weibull distribution.

Examples

dist_w <- dist_weibull(shape = 2, scale = 5)

Parameter Estimation for Hybrid Type-II Censored Data using MleCensoR

Description

Fits process distribution parameters under Hybrid Type-II censored lifetime data using the MleCensoR package (MleCensoR::mle_hybrid_type2) with robust numerical optimization fallbacks.

Usage

fit_distribution_hybrid2(
  x,
  r,
  tc,
  n,
  distribution,
  start = NULL,
  method = NULL,
  lower = NULL,
  upper = NULL,
  ...
)

gpci_fit_hybrid2_dist(
  x,
  r,
  tc,
  n,
  distribution,
  start = NULL,
  method = NULL,
  lower = NULL,
  upper = NULL,
  ...
)

fit_hybrid2(
  x,
  r,
  tc,
  n,
  distribution,
  start = NULL,
  method = NULL,
  lower = NULL,
  upper = NULL,
  ...
)

Arguments

Details

Under Hybrid Type-II censoring, n units are put on life test. The experiment terminates at T^* = \max(x_r, T_c), where r is the pre-fixed target number of failures and T_c is the pre-fixed censoring time. The log-likelihood function is:

\ell(\theta) = \sum_{i=1}^d \ln f(x_i; \theta) + (n - d) \ln S(T^*; \theta)

where d = \mathrm{length}(x) is the number of observed failures up to time T^*, and S(t; \theta) = 1 - F(t; \theta) is the survival function.

Value

A fitted gpci_dist_hybrid2 distribution object with updated parameter estimates and attributes:

Examples

dist_exp <- dist_exponential(rate = 1)
x <- c(0.2, 0.5, 0.8, 1.1)
fit <- fit_distribution_hybrid2(x = x, r = 3, tc = 1.0, n = 10, distribution = dist_exp)
print(fit)

Goodness-of-Fit Testing for Hybrid Type-II Censored Data using gofPHCS

Description

Performs goodness-of-fit testing for Hybrid Type-II censored lifetime data using the gofPHCS package (gofPHCS::gof_test).

Usage

gof_test_hybrid2(
  fit = NULL,
  x = NULL,
  r = NULL,
  tc = NULL,
  n = NULL,
  distribution = NULL,
  statistic = "auto",
  p.method = c("auto", "asymptotic", "montecarlo"),
  nsim = 999,
  seed = NULL,
  conf.level = 0.95,
  ...
)

gof_test(
  fit = NULL,
  x = NULL,
  r = NULL,
  tc = NULL,
  n = NULL,
  distribution = NULL,
  statistic = "auto",
  p.method = c("auto", "asymptotic", "montecarlo"),
  nsim = 999,
  seed = NULL,
  conf.level = 0.95,
  ...
)

Arguments

Details

Goodness-of-fit testing verifies whether the assumed continuous probability distribution fits the observed failure times under Hybrid Type-II censoring. The test utilizes the specialized framework of the 'gofPHCS' package.

Value

An S3 object of class "gpci_gof_hybrid2" containing:

Examples

dist_exp <- dist_exponential(rate = 0.5)
x_data <- c(0.2, 0.5, 0.8, 1.1)
gof_res <- gof_test_hybrid2(
  x = x_data, r = 3, tc = 1.0, n = 10,
  distribution = dist_exp, p.method = "montecarlo", nsim = 50
)
print(gof_res)

Extract a Specific Capability Index

Description

Extract a Specific Capability Index

Usage

gpci_index_hybrid2(fit, index)

Arguments

Value

Numeric scalar value of the index.

Examples

dist_exp <- dist_exponential(rate = 1)
x <- c(0.2, 0.5, 0.8, 1.1)
fit <- capability_hybrid2(x = x, r = 3, tc = 1.0, n = 10, distribution = dist_exp, USL = 3, LSL = 0)
gpci_index_hybrid2(fit, "Cpy")

High-Level User Interface Function for Hybrid Type-II Lindley Approximation GPCI Analysis

Description

Main user-facing function allowing input of custom PDF, CDF, and Survival functions, prior distribution hyperparameters, chain length, burn-in, and thinning under Hybrid Type-II censoring.

Usage

gpci_lindley_hybrid2(
  x,
  r,
  tc,
  n,
  pdf = NULL,
  cdf = NULL,
  sf = NULL,
  quantile = NULL,
  prior = NULL,
  chain_length = 1000,
  burn_in = 200,
  thinning = 1,
  USL,
  LSL,
  target = (USL + LSL)/2,
  indices = NULL,
  B = 500,
  ...
)

gpcihybridIILinApp(
  x,
  r,
  tc,
  n,
  pdf = NULL,
  cdf = NULL,
  sf = NULL,
  quantile = NULL,
  prior = NULL,
  chain_length = 1000,
  burn_in = 200,
  thinning = 1,
  USL,
  LSL,
  target = (USL + LSL)/2,
  indices = NULL,
  B = 500,
  ...
)

Arguments

Value

An object of S3 class "gpcihybridIILinApp" containing MCMC parameter chains, GPCI chains, MLE point estimates, Lindley Bayes point estimates, bootstrap confidence intervals, and analysis specifications.

Examples

my_pdf <- function(x, rate = 1) stats::dexp(x, rate = rate)
my_cdf <- function(x, rate = 1) stats::pexp(x, rate = rate)
my_sf  <- function(x, rate = 1) stats::pexp(x, rate = rate, lower.tail = FALSE)

fit_res <- gpci_lindley_hybrid2(
  x = c(0.2, 0.5, 0.8, 1.1),
  r = 3, tc = 1.0, n = 10,
  pdf = my_pdf, cdf = my_cdf, sf = my_sf,
  chain_length = 20, burn_in = 5,
  USL = 3, LSL = 0,
  indices = c("Cpy", "Cp"),
  B = 5
)
print(fit_res)

Heidelberger and Welch's MCMC Convergence Diagnostic

Description

Conducts stationarity and relative half-width tests under Heidelberger and Welch's MCMC convergence diagnostic, returning test statistics, status, and convergence probability.

Usage

heidelberger_welch(x, alpha = 0.05, eps = 0.1)

Arguments

Details

Implements the two-stage diagnostic procedure of Heidelberger & Welch (1983). First, the Cramér-von Mises statistic is computed on cumulative sums of the series using batch-means spectral density variance estimation at frequency zero. If the test fails, successive 10 achieved. Second, a relative half-width test is performed to verify whether the half-width of the sample mean confidence interval is smaller than eps * |mean|.

Value

A list containing Heidelberger and Welch convergence diagnostic results:

Examples

samples <- stats::rnorm(1000)
heidelberger_welch(samples)

Highest Posterior Density (HPD) Interval Calculation

Description

Computes the Highest Posterior Density (HPD) interval for sample draws at specified confidence / credibility levels (e.g., 90

Usage

hpd_interval(x, prob = 0.95)

Arguments

Value

A named numeric vector of length 2 with elements "lower" and "upper" specifying the Highest Posterior Density (HPD) interval bounds at the requested credibility level.

Examples

samples <- stats::rnorm(1000, mean = 5, sd = 1)
hpd_interval(samples, prob = 0.95)

Lindley Approximation Method for Parameter and GPCI Bayesian Estimation under Hybrid Type-II Censoring

Description

Computes Bayesian posterior expectations of model parameters and Generalized Process Capability Indices (GPCIs) using Lindley's 3rd-order Taylor series approximation for Hybrid Type-II censored lifetime data.

Usage

lindley_approx_hybrid2(
  x,
  r,
  tc,
  n,
  distribution,
  prior = NULL,
  USL,
  LSL,
  target = (USL + LSL)/2,
  indices = NULL,
  u = 1,
  v = 1,
  C0 = 1,
  C1 = 0,
  C2 = 1,
  P0 = 0.9973002,
  ...
)

Arguments

Details

Under Hybrid Type-II censoring with initial sample size n, target failures r, and censoring time T_c, the experiment terminates at T^* = \max(x_r, T_c). The likelihood is:

L(\theta \mid x, r, T_c, n) = \left[ \prod_{i=1}^d f(x_i; \theta) \right] [S(T^*; \theta)]^{n-d}

where d = \mathrm{length}(x) \ge r is the number of observed failures.

Lindley (1980) proposed an asymptotic expansion for the posterior expectation of an arbitrary function u(\theta):

E[u(\theta) \mid \mathrm{data}] \approx u(\hat{\theta}) + \frac{1}{2} \sum_{i,j} (u_{ij} + 2 u_i \rho_j) \sigma_{ij} + \frac{1}{2} \sum_{i,j,k,l} L_{ijk} \sigma_{ij} \sigma_{kl} u_l

where \hat{\theta} is the MLE under Hybrid Type-II censoring, \sigma_{ij} are elements of the asymptotic covariance matrix \Sigma = [-\nabla^2 \ell(\hat{\theta})]^{-1}, \rho_j = \partial \ln \pi(\theta) / \partial \theta_j is the log-prior gradient at \hat{\theta}, and L_{ijk} = \partial^3 \ell / \partial \theta_i \partial \theta_j \partial \theta_k are third-order log-likelihood derivatives evaluated at \hat{\theta}.

Value

An object of S3 class "gpciLindleyHybrid2" containing a list with:

Examples

dist_exp <- dist_exponential(rate = 1)
x_data <- c(0.2, 0.5, 0.8, 1.1)
fit_lnd <- lindley_approx_hybrid2(
  x = x_data, r = 3, tc = 1.0, n = 10,
  distribution = dist_exp, USL = 3, LSL = 0
)
print(fit_lnd)

Lindley Approximation MCMC-Style Chain Generator for GPCIs under Hybrid Type-II Censoring

Description

Generates posterior parameter draws, GPCI chains, point estimates, HPD intervals, bootstrap confidence intervals, and convergence diagnostics for Hybrid Type-II censored data using the Lindley approximation method combined with sampling, burn-in, and thinning.

Usage

lindley_gpci_hybrid2(
  x,
  r,
  tc,
  n,
  distribution = NULL,
  prior = NULL,
  chain_length = 1000,
  burn_in = 200,
  thinning = 1,
  USL,
  LSL,
  target = (USL + LSL)/2,
  indices = NULL,
  u = 1,
  v = 1,
  C0 = 1,
  C1 = 0,
  C2 = 1,
  P0 = 0.9973002,
  B = 500,
  ...
)

Arguments

Details

Under Hybrid Type-II censoring with sample size n, target failures r, and censoring time T_c, the experiment terminates at T^* = \max(x_r, T_c).

The function first computes Maximum Likelihood Estimates (MLE) and Lindley Bayes point estimates of model parameters and GPCIs using 3rd-order Taylor series approximations. Next, parameter draws are generated centered at the Lindley Bayes posterior mode using the asymptotic covariance matrix. Burn-in and thinning are applied to extract the retained parameter chain, which is then mapped through all requested capability indices to generate the posterior GPCI chain. Finally, parametric or non-parametric bootstrap resampling is executed to provide robust confidence intervals.

Value

An object of S3 class "gpcihybridIILinApp" containing a list with the following components:

Examples

dist_exp <- dist_exponential(rate = 1)
x_data <- c(0.2, 0.5, 0.8, 1.1)
fit_chain <- lindley_gpci_hybrid2(
  x = x_data, r = 3, tc = 1.0, n = 10,
  distribution = dist_exp,
  chain_length = 20,
  burn_in = 5,
  thinning = 1,
  USL = 3,
  LSL = 0,
  indices = c("Cpy", "Cp"),
  B = 5
)
print(fit_chain)

Plot Method for gpciBootstrapHybrid2 Objects

Description

Visualizes bootstrap distributions and confidence limits for GPCIs.

Usage

## S3 method for class 'gpciBootstrapHybrid2'
plot(x, indices = NULL, ...)

Arguments

Value

Invisible x.

Examples

dist_exp <- dist_exponential(rate = 1)
x_data <- c(0.2, 0.5, 0.8, 1.1)
boot_res <- bootstrap_hybrid2_gpci(
  x = x_data, r = 3, tc = 1.0, n = 10,
  distribution = dist_exp, USL = 3, LSL = 0, indices = "Cpy", B = 5
)
plot(boot_res, indices = "Cpy")

Plot Method for gpcihybridIILinApp Objects

Description

Visualizes posterior parameter and GPCI chains (trace plots, density plots) generated from Lindley approximation analysis under Hybrid Type-II censoring.

Usage

## S3 method for class 'gpcihybridIILinApp'
plot(x, which = c("both", "trace", "density"), indices = NULL, ...)

Arguments

Value

Invisible x.

Examples

dist_exp <- dist_exponential(rate = 1)
x_data <- c(0.2, 0.5, 0.8, 1.1)
fit_chain <- lindley_gpci_hybrid2(
  x = x_data, r = 3, tc = 1.0, n = 10,
  distribution = dist_exp,
  chain_length = 20, burn_in = 5,
  USL = 3, LSL = 0, indices = "Cpy", B = 5
)
plot(fit_chain, which = "trace", indices = "Cpy")

Print Method for gpciBootstrapHybrid2 Objects

Description

Print Method for gpciBootstrapHybrid2 Objects

Usage

## S3 method for class 'gpciBootstrapHybrid2'
print(x, ...)

Arguments

Value

Invisible x.

Print Method for gpciLindleyHybrid2 Objects

Description

Print Method for gpciLindleyHybrid2 Objects

Usage

## S3 method for class 'gpciLindleyHybrid2'
print(x, ...)

Arguments

Value

Invisible x.

Print Method for gpci_dist_hybrid2

Description

Print Method for gpci_dist_hybrid2

Usage

## S3 method for class 'gpci_dist_hybrid2'
print(x, ...)

Arguments

Value

Invisible x.

Print Method for gpci_gof_hybrid2

Description

Print Method for gpci_gof_hybrid2

Usage

## S3 method for class 'gpci_gof_hybrid2'
print(x, ...)

Arguments

Value

Invisible x.

Print Method for gpcifit_hybrid2

Description

Print Method for gpcifit_hybrid2

Usage

## S3 method for class 'gpcifit_hybrid2'
print(x, ...)

Arguments

Value

Invisible x.

Print Method for gpcihybridIILinApp Objects

Description

Print Method for gpcihybridIILinApp Objects

Usage

## S3 method for class 'gpcihybridIILinApp'
print(x, ...)

Arguments

Value

Invisible x.

Summary Method for gpciBootstrapHybrid2 Objects

Description

Summary Method for gpciBootstrapHybrid2 Objects

Usage

## S3 method for class 'gpciBootstrapHybrid2'
summary(object, ...)

Arguments

Value

Invisible object.

Summary Method for gpciLindleyHybrid2 Objects

Description

Summary Method for gpciLindleyHybrid2 Objects

Usage

## S3 method for class 'gpciLindleyHybrid2'
summary(object, ...)

Arguments

Value

Invisible object.

Summary Method for gpci_dist_hybrid2

Description

Summary Method for gpci_dist_hybrid2

Usage

## S3 method for class 'gpci_dist_hybrid2'
summary(object, ...)

Arguments

Value

Invisible object.

Summary Method for gpci_gof_hybrid2

Description

Summary Method for gpci_gof_hybrid2

Usage

## S3 method for class 'gpci_gof_hybrid2'
summary(object, ...)

Arguments

Value

Invisible object.

Summary Method for gpcifit_hybrid2

Description

Summary Method for gpcifit_hybrid2

Usage

## S3 method for class 'gpcifit_hybrid2'
summary(object, ...)

Arguments

Value

Invisible object.

Summary and Diagnostics for gpcihybridIILinApp Objects

Description

Computes point estimates, Lindley Bayes estimates, Bias, MSE, Risk values (Linex + SEL), HPD intervals at 90 convergence diagnostic, convergence probability, and Bootstrap confidence intervals under Hybrid Type-II censoring.

Usage

## S3 method for class 'gpcihybridIILinApp'
summary(object, ...)

Arguments

Value

A data.frame containing detailed diagnostic metrics for each evaluated Generalized Process Capability Index (GPCI), including MLE estimates, Lindley Bayes estimates, MCMC chain means, bias, mean squared error (MSE), risk values (SEL + Linex), 90%, 95%, and 99% HPD credible intervals, Heidelberger-Welch stationarity test statistics and p-values, convergence probabilities, and 95% bootstrap percentile confidence intervals.

Examples

dist_exp <- dist_exponential(rate = 1)
x_data <- c(0.2, 0.5, 0.8, 1.1)
fit_chain <- lindley_gpci_hybrid2(
  x = x_data, r = 3, tc = 1.0, n = 10,
  distribution = dist_exp,
  USL = 3, LSL = 0, chain_length = 20, burn_in = 5,
  indices = c("Cpy", "Cp"), B = 5
)
summary(fit_chain)

Vcov Method for gpcifit_hybrid2

Description

Extract asymptotic variance-covariance matrix of fitted distribution parameters.

Usage

## S3 method for class 'gpcifit_hybrid2'
vcov(object, ...)

Arguments

Value

Variance-covariance matrix.