Package {gpcihybridIIImpSam}

Type:	Package
Title:	Process Capability Indices for Hybrid Type-II Data via Importance Sampling
Version:	0.1.0
Maintainer:	Shikhar Tyagi <shikhar1093tyagi@gmail.com>
Description:	Evaluates Generalized Process Capability Indices (GPCIs) under Hybrid Type-II censored lifetime data using Importance Sampling (Sampling Importance Resampling, SIR). Implements Bayesian parameter estimation and evaluates classical and generalized capability indices including Cpy, Cp, Cpk, Cpu, Cpl, Cpm, Cpmk, Spmk, CpTk, Cpc, CNp, CNpk, CNpm, CNpmk, CNpmc, CNpmkc, and Vannman's Cp(u,v) family. Computes initial maximum likelihood estimates under Hybrid Type-II censoring, parameter MCMC chains, GPCI posterior chains, posterior point estimates, bias, mean squared error (MSE), Bayes risk, Highest Posterior Density (HPD) credible intervals at 90%, 95%, and 99% levels, Heidelberger and Welch's MCMC convergence diagnostics, and convergence probabilities. Accommodates user-defined probability density/mass functions, cumulative distribution functions, and survival functions. Goodness-of-fit testing for Hybrid Type-II censored data is supported via 'gofPHCS'. Methods are based on Childs et al. (2003) <doi:10.1080/0266476032000053637>, Kundu and Pradhan (2009) <doi:10.1016/j.spl.2008.09.006>, Maiti et al. (2010) <doi:10.1080/16843703.2010.11673233>, Dey and Saha (2019) <doi:10.1007/s41872-019-00081-4>, Alotaibi et al. (2022) <doi:10.1155/2022/3135264>, Saha et al. (2022) <doi:10.1080/02664763.2021.1971632>, and Saha et al. (2024) <doi:10.1142/S021853932450013X>.
License:	GPL-2 | GPL-3 [expanded from: GPL (≥ 2)]
Encoding:	UTF-8
RoxygenNote:	7.3.3
Depends:	R (≥ 4.0.0)
Imports:	coda, stats, graphics
Suggests:	gofPHCS, testthat (≥ 3.0.0), knitr, rmarkdown
VignetteBuilder:	knitr
Config/testthat/edition:	3
NeedsCompilation:	no
Packaged:	2026-08-16 17:34:19 UTC; shikhar tyagi
Author:	Shikhar Tyagi [aut, cre]
Repository:	CRAN
Date/Publication:	2026-08-21 13:00:26 UTC

gpcihybridIIImpSam: Process Capability Indices for Hybrid Type-II Data via Importance Sampling

Description

Evaluates Generalized Process Capability Indices (GPCIs) under Hybrid Type-II censored lifetime data using Importance Sampling (Sampling Importance Resampling, SIR). Implements Bayesian parameter estimation and evaluates classical and generalized capability indices including Cpy, Cp, Cpk, Cpu, Cpl, Cpm, Cpmk, Spmk, CpTk, Cpc, CNp, CNpk, CNpm, CNpmk, CNpmc, CNpmkc, and Vannman's Cp(u,v) family. Computes initial maximum likelihood estimates under Hybrid Type-II censoring, parameter MCMC chains, GPCI posterior chains, posterior point estimates, bias, mean squared error (MSE), Bayes risk, Highest Posterior Density (HPD) credible intervals at 90

Author(s)

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

Coef Method for gpcihybridIIImpSam Objects

Description

Extracts posterior mean estimates of capability indices or distribution parameters.

Usage

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

Arguments

Value

Named numeric vector of posterior estimates.

Compute Generalized Process Capability Indices (GPCIs)

Description

Evaluates classical and generalized Process Capability Indices (GPCIs) given process specification limits and a target distribution specification.

Usage

compute_gpcis(
  distribution,
  USL,
  LSL,
  target = (USL + LSL)/2,
  indices = c("Cpy", "Cp", "Cpk", "Cpm", "Cpmk", "Spmk", "CNpmc"),
  mode = c("moments", "quantile"),
  u = 1,
  v = 1,
  C0 = 1,
  C1 = 0,
  C2 = 1,
  tolerance_t = USL - LSL,
  P0 = 0.9973002,
  LDL = LSL,
  UDL = USL
)

capability(
  distribution,
  USL,
  LSL,
  target = (USL + LSL)/2,
  indices = c("Cpy", "Cp", "Cpk", "Cpm", "Cpmk", "Spmk", "CNpmc"),
  mode = c("moments", "quantile"),
  u = 1,
  v = 1,
  C0 = 1,
  C1 = 0,
  C2 = 1,
  tolerance_t = USL - LSL,
  P0 = 0.9973002,
  LDL = LSL,
  UDL = USL
)

Arguments

Details

Supports moment-based and quantile-based capability indices including:

• C_{py}: Conformance yield capability index (F(USL) - F(LSL)) / P_0.

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

• S_{pmk}: Conformance loss and target deviation index.

• C_{pTk}: Asymmetric target-based index.

• C_{pc}: Cost-ratio capability index.

• C_{Np}, C_{Npk}, C_{Npm}, C_{Npmk}: Quantile-based non-normal indices.

• C_{Npmc}, C_{Npmkc}: Tolerance cost-adjusted capability indices using C_M(t) = C_0 + C_1 \exp(-C_2 t).

• C_{p}(u, v), C_{Np}(u, v): Generalized weighted family of capability indices.

Value

A named numeric vector of computed GPCI values.

Examples

dist <- dist_weibull(shape = 2, scale = 10)
compute_gpcis(dist, USL = 15, LSL = 5, target = 10)

Confint Method for gpcihybridIIImpSam Objects

Description

Extracts HPD credible intervals for capability indices or parameters at 90%, 95%, or 99% level.

Usage

## S3 method for class 'gpcihybridIIImpSam'
confint(
  object,
  parm = NULL,
  level = 0.95,
  what = c("indices", "parameters"),
  ...
)

Arguments

Value

A matrix with columns for the lower and upper bounds of the credible intervals.

Define a Lifetime Distribution Object for Hybrid Type-II Censored Data

Description

Constructor to define a continuous or discrete lifetime distribution object for Bayesian Capability Analysis under Hybrid Type-II Censoring via Importance Sampling.

Usage

define_distribution_hybrid2impsam(
  name = "Custom",
  pdf = NULL,
  cdf = NULL,
  sf = NULL,
  surv = NULL,
  quantile = NULL,
  moments = NULL,
  param_names = NULL,
  default_params = list(),
  params = list(),
  support = c(0, Inf)
)

create_custom_dist(
  name = "Custom",
  pdf = NULL,
  cdf = NULL,
  sf = NULL,
  surv = NULL,
  quantile = NULL,
  moments = NULL,
  param_names = NULL,
  default_params = list(),
  params = list(),
  support = c(0, Inf)
)

Arguments

Details

Users can define a distribution by specifying any of its Probability Density/Mass Function (PDF/PMF), Cumulative Distribution Function (CDF), Survival Function (SF), Quantile Function, or Moments. Missing functions are automatically and stably derived numerically.

Value

An S3 object of class "gpci_dist_hybrid2impsam" and "gpci_dist" containing:

Examples

dist_custom <- define_distribution_hybrid2impsam(
  name = "CustomExp",
  pdf = function(x, rate) stats::dexp(x, rate = rate),
  cdf = function(q, rate) stats::pexp(q, rate = rate),
  param_names = "rate",
  params = list(rate = 1)
)
print(dist_custom)

Exponential Distribution Constructor

Description

Exponential Distribution Constructor

Usage

dist_exponential(rate = 1)

Arguments

Value

A gpci_dist_hybrid2impsam 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_hybrid2impsam object for Gamma distribution.

Examples

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

Lindley Distribution Constructor

Description

Lindley Distribution Constructor

Usage

dist_lindley(theta = 1)

Arguments

Value

A gpci_dist_hybrid2impsam object for Lindley distribution.

Examples

dist_lind <- dist_lindley(theta = 1.5)

Logistic Distribution Constructor

Description

Logistic Distribution Constructor

Usage

dist_logistic(location = 0, scale = 1)

Arguments

Value

A gpci_dist_hybrid2impsam object for Logistic distribution.

Examples

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

Logistic-Exponential Distribution Constructor

Description

Logistic-Exponential Distribution Constructor

Usage

dist_logistic_exponential(alpha = 1, lambda = 1)

Arguments

Value

A gpci_dist_hybrid2impsam object for Logistic-Exponential distribution.

Examples

dist_le <- dist_logistic_exponential(alpha = 1.2, lambda = 0.8)

Log-Logistic Distribution Constructor

Description

Log-Logistic Distribution Constructor

Usage

dist_loglogistic(shape = 1, scale = 1)

Arguments

Value

A gpci_dist_hybrid2impsam 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_hybrid2impsam 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_hybrid2impsam object for Normal distribution.

Examples

dist_n <- 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_hybrid2impsam object for Weibull distribution.

Examples

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

Safe Evaluation of Log-Prior Density

Description

Evaluates the joint log-prior density for distribution parameters.

Usage

eval_log_prior(params, priors = NULL)

Arguments

Value

Numeric scalar representing the joint log-prior density. Returns -Inf for values outside parameter support.

Examples

eval_log_prior(c(shape = 2, scale = 5))
pr_spec <- list(rate = list(dist = "gamma", params = c(shape = 1, rate = 1)))
eval_log_prior(c(rate = 0.5), priors = pr_spec)

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

Value

An S3 object of class "gpci_gof_hybrid2" containing:

Examples

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

Bayesian Importance Sampling Estimation of GPCIs under Hybrid Type-II Censoring

Description

Evaluates Generalized Process Capability Indices (GPCIs) under Hybrid Type-II censored lifetime data using Importance Sampling (Sampling Importance Resampling, SIR).

Usage

gpci_hybrid2_impsam(
  x,
  r,
  tc,
  n,
  distribution = NULL,
  pdf = NULL,
  cdf = NULL,
  surv = NULL,
  quantile = NULL,
  param_names = NULL,
  start = NULL,
  priors = NULL,
  chain_length = 1000,
  burn_in = 200,
  thinning = 1,
  USL,
  LSL,
  target = (USL + LSL)/2,
  indices = c("Cpy", "Cp", "Cpk", "Cpm", "Cpmk", "Spmk", "CNpmc"),
  mode = c("moments", "quantile"),
  u = 1,
  v = 1,
  C0 = 1,
  C1 = 0,
  C2 = 1,
  tolerance_t = USL - LSL,
  P0 = 0.9973002,
  true_param = NULL,
  true_gpci = NULL
)

gpcihybridIIImpSam(
  x,
  r,
  tc,
  n,
  distribution = NULL,
  pdf = NULL,
  cdf = NULL,
  surv = NULL,
  quantile = NULL,
  param_names = NULL,
  start = NULL,
  priors = NULL,
  chain_length = 1000,
  burn_in = 200,
  thinning = 1,
  USL,
  LSL,
  target = (USL + LSL)/2,
  indices = c("Cpy", "Cp", "Cpk", "Cpm", "Cpmk", "Spmk", "CNpmc"),
  mode = c("moments", "quantile"),
  u = 1,
  v = 1,
  C0 = 1,
  C1 = 0,
  C2 = 1,
  tolerance_t = USL - LSL,
  P0 = 0.9973002,
  true_param = NULL,
  true_gpci = NULL
)

Arguments

Details

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

The user can supply observed data (x, r, tc, n), custom functions for probability density/mass (pdf), cumulative distribution (cdf), and survival (surv), along with prior distributions, chain length, burn-in period, and thinning factor.

The algorithm first computes initial parameter estimates under Hybrid Type-II censoring, evaluates baseline GPCI point estimates, runs Importance Sampling (SIR), computes the thinned GPCI chain, and returns comprehensive posterior point estimates, bias, MSE, Bayes risk, HPD credible intervals at 90%, 95%, and 99% levels, Heidelberger and Welch's MCMC convergence diagnostics, and coverage probabilities.

Missing values (NA, NaN) in input vectors or invalid parameter evaluations are safely trapped and handled without errors. Logical inputs are handled as standard boolean values (TRUE or FALSE).

Value

An object of class "gpcihybridIIImpSam" containing:

Examples

# Example using custom user-supplied functions for Exponential lifetime data
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)

x_data <- c(0.5, 1.2, 2.1, 3.4)
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
)
print(fit)

Heidelberger and Welch's MCMC Convergence Diagnostic

Description

Conducts the stationarity test and relative half-width test under Heidelberger and Welch's MCMC convergence diagnostic.

Usage

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

Arguments

Value

A list containing:

Examples

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

Highest Posterior Density (HPD) Interval Calculation

Description

Computes the Highest Posterior Density (HPD) interval for MCMC posterior samples at specified confidence / credibility levels (e.g. 90%, 95%, 99%).

Usage

hpd_interval(x, prob = 0.95)

Arguments

Value

A named numeric vector of length 2 containing lower and upper HPD limits.

Examples

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

Importance Sampling for Hybrid Type-II Censored Data

Description

Fits initial parameter estimates and generates MCMC posterior parameter draws using Importance Sampling (Sampling Importance Resampling, SIR) under Hybrid Type-II censored lifetime data.

Usage

impsam_hybrid_ty2(
  x,
  r,
  tc,
  n,
  distribution,
  start = NULL,
  priors = NULL,
  chain_length = 1000,
  burn_in = 200,
  thinning = 1
)

Arguments

Details

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] \left[ S(T^*; \theta) \right]^{n - d}

where d \ge r is the number of observed failures up to time T^*.

The algorithm first determines the maximum likelihood estimates and Laplace-approximated posterior mode and covariance matrix, draws candidate vectors from a proposal distribution, calculates importance weights, and performs Sampling Importance Resampling (SIR).

Value

A list containing:

Examples

dist_exp <- dist_exponential(rate = 1)
x_obs <- c(0.4, 0.9, 1.5, 2.3)
res <- impsam_hybrid_ty2(x = x_obs, r = 3, tc = 2.0, n = 10, distribution = dist_exp,
                         chain_length = 500, burn_in = 100, thinning = 1)

Plot Method for gpcihybridIIImpSam Objects

Description

Generates posterior trace plots and density histograms with Highest Posterior Density (HPD) intervals.

Usage

## S3 method for class 'gpcihybridIIImpSam'
plot(x, index = NULL, ...)

Arguments

Value

Invisibly returns the input object x.

Print Method for gpci_dist_hybrid2impsam Objects

Description

Print Method for gpci_dist_hybrid2impsam Objects

Usage

## S3 method for class 'gpci_dist_hybrid2impsam'
print(x, ...)

Arguments

Value

The invisible object x.

Print Method for gpci_gof_hybrid2 Objects

Description

Print Method for gpci_gof_hybrid2 Objects

Usage

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

Arguments

Value

The invisible object x.

Print Method for gpcihybridIIImpSam Objects

Description

Prints a structured summary of initial point estimates and MCMC posterior capability analysis results.

Usage

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

Arguments

Value

Invisibly returns the input object x.

Print Method for summary.gpcihybridIIImpSam Objects

Description

Print Method for summary.gpcihybridIIImpSam Objects

Usage

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

Arguments

Value

Invisibly returns the input object x.

Summarize MCMC Posterior Samples for Parameters and Capability Indices

Description

Computes posterior mean point estimate, bias, mean squared error (MSE), Bayes risk, Highest Posterior Density (HPD) intervals at 90%, 95%, and 99% levels, Heidelberger-Welch convergence diagnostics, and empirical coverage probabilities.

Usage

summarize_chain(chain, true_val = NULL)

Arguments

Value

A data frame containing:

Examples

samples <- stats::rnorm(1000, mean = 1.5, sd = 0.2)
summarize_chain(samples, true_val = 1.5)

Summary Method for gpci_dist_hybrid2impsam Objects

Description

Summary Method for gpci_dist_hybrid2impsam Objects

Usage

## S3 method for class 'gpci_dist_hybrid2impsam'
summary(object, ...)

Arguments

Value

The invisible object object.

Summary Method for gpci_gof_hybrid2 Objects

Description

Summary Method for gpci_gof_hybrid2 Objects

Usage

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

Arguments

Value

The invisible object object.

Summary Method for gpcihybridIIImpSam Objects

Description

Provides complete posterior statistical summary including HPD intervals and Heidelberger-Welch diagnostics.

Usage

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

Arguments

Value

An S3 object of class "summary.gpcihybridIIImpSam" containing parameter and GPCI summary tables.

Validate Hybrid Type-II Censored Data Inputs

Description

Validates and formats inputs for Hybrid Type-II censored lifetime data.

Usage

validate_hybrid2_data(x, r, tc, n)

Arguments

Details

Under Hybrid Type-II censoring, a total of n items are put on life test. The experiment terminates at time 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. At least r failures must be observed (d \ge r).

Value

A validated list containing:

Examples

validate_hybrid2_data(x = c(0.5, 1.2, 2.1, 3.4), r = 3, tc = 2.5, n = 10)