Package {gpciEMprogII}

Type: Package
Title: Generalized Process Capability Indices via EM Algorithm under Progressive Type-II Censoring
Version: 0.1.0
Description: Implements the Expectation-Maximization (EM) algorithm of Dempster, Laird, and Rubin (1977) <doi:10.1111/j.2517-6161.1977.tb01600.x> for parameter estimation under progressive Type-II censored data (Balakrishnan and Aggarwala (2000) <doi:10.1007/978-1-4612-1186-0>) and computes Generalized Process Capability Indices (GPCIs). Uses the 'UniCensorEM' package for EM estimation. Supports classical and generalized capability indices including Cpy (Maiti et al. (2010) <doi:10.1080/16843703.2010.11673233>), Cp, Cpk, Cpu, Cpl, Cpm, Cpmk, Spmk (Dey and Saha (2019) <doi:10.1007/s41872-019-00081-4>), CpTk (Saha et al. (2018) <doi:10.1080/21681015.2018.1437793>), Cpc, CNpmc (Alotaibi et al. (2022) <doi:10.1155/2022/3135264>), CNpmkc (Saha et al. (2024) <doi:10.1142/S021853932450013X>), and CNpk (Saha et al. (2022) <doi:10.1080/02664763.2021.1971632>). Computes point estimates, bias, mean squared error, risk, Heidelberger and Welch convergence diagnostics, convergence probability, and bootstrap confidence intervals at 90 percent, 95 percent, and 99 percent levels. Accommodates user-defined probability density or mass functions, cumulative distribution functions, and survival functions.
License: MIT + file LICENSE
Encoding: UTF-8
RoxygenNote: 7.3.3
Depends: R (≥ 4.0.0)
Imports: stats, UniCensorEM
Suggests: testthat (≥ 3.0.0)
Config/testthat/edition: 3
NeedsCompilation: no
Packaged: 2026-08-03 04:02:24 UTC; shikhar tyagi
Author: Shikhar Tyagi ORCID iD [aut, cre]
Maintainer: Shikhar Tyagi <shikhar1093tyagi@gmail.com>
Repository: CRAN
Date/Publication: 2026-08-08 12:50:02 UTC

Bootstrap Confidence Intervals for Progressive Type-II GPCI (EM)

Description

Computes parametric or non-parametric bootstrap confidence intervals for Generalized Process Capability Indices and model parameters under progressive Type-II censoring using EM algorithm estimation. Provides bootstrap CIs at 90%, 95%, and 99% levels, plus estimates of bias, MSE, risk, Heidelberger-Welch convergence diagnostics, and convergence probability.

Usage

boot_ci_prog_em(
  fit,
  B = 1000,
  alpha = c(0.1, 0.05, 0.01),
  method = c("percentile", "normal", "basic", "BCp", "BCa", "studentized"),
  type = c("parametric", "nonparametric"),
  max_iter = 1000,
  tol = 1e-08
)

gpc_boot_em(
  fit,
  B = 1000,
  alpha = c(0.1, 0.05, 0.01),
  method = c("percentile", "normal", "basic", "BCp", "BCa", "studentized"),
  type = c("parametric", "nonparametric"),
  max_iter = 1000,
  tol = 1e-08
)

Arguments

fit

A gpc_em_fit object returned by capability_prog_em.

B

Number of bootstrap replicates (default 1000).

alpha

Numeric vector of significance levels (default c(0.10, 0.05, 0.01) for 90%, 95%, 99% CIs).

method

CI method: "percentile", "normal", "basic", "BCp", "BCa", or "studentized".

type

Resampling type: "parametric" (default) or "nonparametric".

max_iter

Maximum EM iterations per bootstrap (default 1000).

tol

Convergence tolerance per bootstrap (default 1e-8).

Value

An object of class "gpc_em_ci" containing:

ci_table

Data frame of confidence intervals.

param_perf

Data frame of parameter performance metrics (estimate, SE, bias, MSE, risk).

index_perf

Data frame of index performance metrics.

convergence

Data frame with Heidelberger-Welch diagnostics and convergence probability for each index.

param_reps

Matrix of bootstrap parameter replicates.

index_reps

Matrix of bootstrap index replicates.

fit

Original fitted object.

B

Number of bootstrap replicates.

Examples

my_dist <- dist_weibull(shape = 1, scale = 1)
x <- c(0.5, 1.2, 2.1, 3.4, 4.8)
r <- c(1, 0, 2, 0, 1)
fit <- capability_prog_em(x = x, r_removals = r,
  distribution = my_dist, USL = 6, LSL = 0)
ci <- boot_ci_prog_em(fit, B = 100,
  alpha = c(0.10, 0.05, 0.01), method = "percentile")
print(ci)

Compute Generalized Process Capability Indices via EM Algorithm

Description

Computes classical and generalized Process Capability Indices (PCIs) for progressive Type-II censored data using model parameter estimates obtained from the EM algorithm.

Usage

capability_prog_em(
  x = NULL,
  r_removals = NULL,
  distribution,
  USL,
  LSL,
  target = (USL + LSL)/2,
  indices = c("Cpy", "Cp", "Cpk", "Cpu", "Cpl", "Cpm", "Cpmk", "Spmk", "CpTk", "Cpc",
    "CNpmc"),
  u = 1,
  v = 1,
  mode = c("moments", "quantile"),
  fit = TRUE,
  C0 = 1,
  C1 = 0,
  C2 = 1,
  tolerance_t = USL - LSL,
  P0 = 0.9973002,
  LDL = LSL,
  UDL = USL,
  max_iter = 1000,
  tol = 1e-08
)

gpc_em_fit(
  x = NULL,
  r_removals = NULL,
  distribution,
  USL,
  LSL,
  target = (USL + LSL)/2,
  indices = c("Cpy", "Cp", "Cpk", "Cpu", "Cpl", "Cpm", "Cpmk", "Spmk", "CpTk", "Cpc",
    "CNpmc"),
  u = 1,
  v = 1,
  mode = c("moments", "quantile"),
  fit = TRUE,
  C0 = 1,
  C1 = 0,
  C2 = 1,
  tolerance_t = USL - LSL,
  P0 = 0.9973002,
  LDL = LSL,
  UDL = USL,
  max_iter = 1000,
  tol = 1e-08
)

Arguments

x

Numeric vector of observed failure times under progressive Type-II censoring. Can be NULL if parameters are already fitted.

r_removals

Numeric vector of removal counts (censoring scheme R).

distribution

A gpc_dist distribution object.

USL

Upper Specification Limit.

LSL

Lower Specification Limit.

target

Process target (defaults to (USL + LSL) / 2).

indices

Character vector of capability indices to compute. Choices: "Cpy", "Cp", "Cpk", "Cpu", "Cpl", "Cpm", "Cpmk", "Cp_uv", "Spmk", "CpTk", "Cpc", "CNpmc", "CNpmkc", "CNpk", "Cp_q", "Cpk_q".

u

Weight parameter for Cp(u,v) family (default 1).

v

Weight parameter for Cp(u,v) family (default 1).

mode

Mode of evaluation: "moments" (mean and variance) or "quantile" (quantile-based).

fit

Logical. If TRUE (default), estimates distribution parameters using the EM algorithm before computing indices.

C0

Cost function parameter (default 1).

C1

Cost function parameter (default 0).

C2

Cost function parameter (default 1).

tolerance_t

Tolerance length (defaults to USL - LSL).

P0

Desired conformance level for Cpy and Cpc (default 0.9973002).

LDL

Lower Desired Limit for CpTk (defaults to LSL).

UDL

Upper Desired Limit for CpTk (defaults to USL).

max_iter

Maximum EM iterations (default 1000).

tol

Convergence tolerance for EM (default 1e-8).

Value

An object of class c("gpc_em_fit", "gpcifit") containing:

x

Observed data (if provided).

r_removals

Removal scheme (if provided).

distribution

Fitted distribution object.

USL, LSL, target

Specification limits and target.

indices

Names of computed indices.

estimates

Named numeric vector of index values.

p_hat

Estimated nonconforming proportion.

mode

Computation mode used.

options

List of all computation options.

Examples

my_dist <- dist_weibull(shape = 1, scale = 1)
x <- c(0.5, 1.2, 2.1, 3.4, 4.8)
r <- c(1, 0, 2, 0, 1)
cap <- capability_prog_em(x = x, r_removals = r,
  distribution = my_dist, USL = 6, LSL = 0)
print(cap)

Coef Method for Progressive Type-II GPCI EM Fit

Description

Coef Method for Progressive Type-II GPCI EM Fit

Usage

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

Arguments

object

Object of class gpc_em_fit.

what

Extract "indices" (default) or "parameters".

...

Additional arguments (currently unused).

Value

Named numeric vector of index values or parameter estimates.

Monte Carlo Simulation and Coverage Probability Evaluation

Description

Evaluates standard error, MSE, bias, risk, and empirical coverage probability for parameters and GPCIs under progressive Type-II censoring using the EM algorithm with bootstrap CIs.

Usage

eval_performance_em(
  distribution,
  r_removals,
  USL,
  LSL,
  target = (USL + LSL)/2,
  true_indices = NULL,
  M = 100,
  B = 200,
  alpha = c(0.1, 0.05, 0.01),
  max_iter = 1000,
  tol = 1e-08
)

gpc_sim_em(
  distribution,
  r_removals,
  USL,
  LSL,
  target = (USL + LSL)/2,
  true_indices = NULL,
  M = 100,
  B = 200,
  alpha = c(0.1, 0.05, 0.01),
  max_iter = 1000,
  tol = 1e-08
)

Arguments

distribution

A gpc_dist distribution object with true parameter values.

r_removals

Progressive censoring scheme vector R.

USL

Upper Specification Limit.

LSL

Lower Specification Limit.

target

Process target (default midpoint).

true_indices

Optional named vector of true index values.

M

Number of Monte Carlo simulation runs (default 100).

B

Number of bootstrap iterations per run (default 200).

alpha

Significance levels (default c(0.10, 0.05, 0.01)).

max_iter

Maximum EM iterations (default 1000).

tol

Convergence tolerance (default 1e-8).

Value

A list containing coverage probabilities, M, B, true parameters, and true index values.

Examples

dist_true <- dist_weibull(shape = 2, scale = 5)
r_scheme <- c(2, 0, 3, 0, 1)
eval_res <- eval_performance_em(dist_true, r_scheme,
  USL = 10, LSL = 0, M = 5, B = 20)
print(eval_res)

Parameter Estimation via EM Algorithm for Progressive Type-II Censored Data

Description

Fits process distribution parameters to progressive Type-II censored data using the Expectation-Maximization (EM) algorithm via the UniCensorEM package.

Usage

fit_prog_ty2_em(
  x,
  r_removals,
  distribution,
  start = NULL,
  max_iter = 1000,
  tol = 1e-08,
  ...
)

Arguments

x

Numeric vector of observed order statistics (sorted ascending).

r_removals

Integer vector of progressive removal counts (censoring scheme R). Must have the same length as x.

distribution

A gpc_dist distribution object created by make_gpc_dist or a predefined distribution constructor.

start

Optional vector of initial parameter values. If NULL, the default parameters from the distribution object are used.

max_iter

Maximum number of EM iterations (default 1000).

tol

Convergence tolerance (default 1e-8).

...

Additional arguments (currently unused).

Value

The input distribution object with updated parameter estimates and attributes: "em_fit" (the raw emfit object), "converged" (logical), "iterations" (integer), and "loglik" (numeric).

Examples

my_dist <- dist_weibull(shape = 1, scale = 1)
x <- c(0.5, 1.2, 2.1, 3.4, 4.8)
r <- c(1, 0, 2, 0, 1)
fitted <- fit_prog_ty2_em(x = x, r_removals = r, distribution = my_dist)

Retrieve a Single GPCI Value

Description

Extracts or computes a single capability index from a fitted object.

Usage

gpc_index_em(fit, index)

Arguments

fit

A gpc_em_fit object.

index

Character name of the index to extract or compute.

Value

Numeric value of the specified index.

Examples

my_dist <- dist_weibull(shape = 1, scale = 1)
x <- c(0.5, 1.2, 2.1, 3.4, 4.8)
r <- c(1, 0, 2, 0, 1)
fit <- capability_prog_em(x = x, r_removals = r,
  distribution = my_dist, USL = 6, LSL = 0, fit = FALSE)
gpc_index_em(fit, "Cp")

Custom and Predefined Distribution Construction

Description

Builds a distribution specification object for parameter estimation using the EM algorithm and Generalized Process Capability Indices (GPCIs) under progressive Type-II censored data.

Usage

make_gpc_dist(
  name,
  pdf,
  cdf,
  surv = NULL,
  quantile = NULL,
  start,
  param_names = paste0("param", seq_along(start)),
  support = c(0, Inf)
)

Arguments

name

Character string giving the name of the distribution.

pdf

Function f(x, theta) for the probability density function (or probability mass function).

cdf

Function F(x, theta) for the cumulative distribution function.

surv

Function S(x, theta) for the survival function (defaults to 1 - cdf(x, theta)).

quantile

Optional function Q(p, theta) for the quantile function. If NULL, quantiles are computed by numerical inversion of the CDF.

start

Vector of initial parameter values for estimation.

param_names

Character vector of parameter names.

support

Numeric vector of length 2 giving the domain support of the distribution (default c(0, Inf)).

Value

An object of class "gpc_dist".

Examples

pdf_exp <- function(x, theta) dexp(x, rate = theta[1])
cdf_exp <- function(x, theta) pexp(x, rate = theta[1])
surv_exp <- function(x, theta) 1 - pexp(x, rate = theta[1])
my_dist <- make_gpc_dist("Exponential", pdf_exp, cdf_exp, surv_exp,
  start = c(1), param_names = "rate")

Predefined Distribution Specifications

Description

Built-in helper functions creating "gpc_dist" objects for standard process distributions.

Usage

dist_normal(mean = 0, sd = 1)

dist_weibull(shape = 1, scale = 1)

dist_gamma(shape = 1, rate = 1)

dist_exp_exp(alpha = 1, lambda = 1)

dist_logistic_exp(alpha = 1, lambda = 1)

Arguments

mean

Initial mean for Normal distribution.

sd

Initial standard deviation for Normal distribution.

shape

Initial shape parameter.

scale

Initial scale parameter.

rate

Initial rate parameter.

alpha

Initial alpha shape parameter.

lambda

Initial lambda scale parameter.

Value

A "gpc_dist" object.

Examples

d_norm <- dist_normal()
d_weib <- dist_weibull()

Print Method for Bootstrap CIs

Description

Print Method for Bootstrap CIs

Usage

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

Arguments

x

Object of class gpc_em_ci.

...

Additional arguments (currently unused).

Value

The input object, invisibly.

Print Method for Progressive Type-II GPCI EM Fit

Description

Print Method for Progressive Type-II GPCI EM Fit

Usage

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

Arguments

x

Object of class gpc_em_fit.

...

Additional arguments (currently unused).

Value

The input object, invisibly.

Summary Method for Bootstrap CIs

Description

Summary Method for Bootstrap CIs

Usage

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

Arguments

object

Object of class gpc_em_ci.

...

Additional arguments (currently unused).

Value

The input object, invisibly.

Summary Method for Progressive Type-II GPCI EM Fit

Description

Summary Method for Progressive Type-II GPCI EM Fit

Usage

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

Arguments

object

Object of class gpc_em_fit.

...

Additional arguments (currently unused).

Value

The input object, invisibly.