Package 'gpciLindleyApprox'

Parametric and Non-Parametric Bootstrap Confidence Intervals for GPCIs

Description

Computes bootstrap confidence intervals for Generalized Process Capability Indices (GPCIs) using parametric or non-parametric resampling. Supports Percentile, Normal, Basic, BCa, BCp, and Studentized bootstrap methods at 90%, 95%, and 99% confidence levels.

Usage

bootstrap_gpci(
  data,
  distribution = NULL,
  USL,
  LSL,
  target = (USL + LSL)/2,
  indices = NULL,
  type = c("nonparametric", "parametric"),
  B = 1000,
  conf_levels = c(0.9, 0.95, 0.99),
  u = 1,
  v = 1
)

Arguments

data	Numeric vector of process observations.
distribution	A gpci_dist object.
USL	Upper Specification Limit.
LSL	Lower Specification Limit.
target	Process target value (default (USL + LSL) / 2).
indices	Character vector of GPCI names to evaluate.
type	Type of bootstrap: "nonparametric" (default) or "parametric".
B	Number of bootstrap replications. Default is 1000.
conf_levels	Confidence levels for intervals (default c(0.90, 0.95, 0.99)).
u, v	Parameters for Cp(u,v) family (default 1).

Value

A list containing bootstrap point estimates, standard errors, and confidence intervals for each index.

Examples

data_x <- stats::rnorm(40, mean = 10, sd = 1)
fit_boot <- bootstrap_gpci(data = data_x, distribution = dist_normal(),
                          USL = 13, LSL = 7, B = 200)

---

Compute Generalized Process Capability Indices (GPCIs)

Description

Computes Generalized Process Capability Indices (GPCIs) for uncensored process data, given a gpci_dist object or data and specification limits.

Usage

capability(
  data = NULL,
  distribution = NULL,
  USL,
  LSL,
  target = (USL + LSL)/2,
  indices = NULL,
  fit = TRUE,
  fit_method = "mle",
  u = 1,
  v = 1,
  C0 = 1,
  C1 = 0,
  C2 = 1,
  P0 = 0.9973002
)

Arguments

data	Numeric vector of uncensored process observations. Optional if distribution has parameters set.
distribution	A gpci_dist object with parameter values.
USL	Upper Specification Limit.
LSL	Lower Specification Limit.
target	Target process value. Default is (USL + LSL) / 2.
indices	Character vector of GPCI names to calculate. Default calculates all available indices.
fit	Logical. If TRUE, fits the distribution to data first. Default TRUE.
fit_method	Method used for distribution fitting (default "mle").
u	Parameter u for Vännman's Cp(u,v) index (default 1).
v	Parameter v for Vännman's Cp(u,v) index (default 1).
C0	Parameter C0 for loss function (default 1).
C1	Parameter C1 for loss function (default 0).
C2	Parameter C2 for loss function (default 1).
P0	Baseline process yield (default 0.9973002).

Value

A named numeric vector of computed GPCI values.

Examples

data_x <- stats::rnorm(50, mean = 10, sd = 1)
dist_norm <- dist_normal()
capability(data = data_x, distribution = dist_norm, USL = 13, LSL = 7)

---

Define a Process Probability Distribution Object

Description

Creates a gpci_dist object containing probability density function (PDF), cumulative distribution function (CDF), survival function (SF), and optional quantile function (QF).

Usage

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

Arguments

name	Character string specifying the name of the distribution.
pdf	Function representing probability density function(x, ...).
cdf	Function representing cumulative distribution function(x, ...).
sf	Function representing survival function function(x, ...).
qf	Optional function representing quantile function function(p, ...).
support	Numeric vector of length 2 giving lower and upper support bounds. Default c(-Inf, Inf).
params	List of initial parameter values or parameter names.

Value

An object of class "gpci_dist".

Examples

dist_norm <- define_distribution(
  name = "Normal",
  pdf = function(x, mean = 0, sd = 1) stats::dnorm(x, mean, sd),
  cdf = function(x, mean = 0, sd = 1) stats::pnorm(x, mean, sd),
  sf  = function(x, mean = 0, sd = 1) 1 - stats::pnorm(x, mean, sd),
  params = list(mean = 0, sd = 1)
)

---

Built-in Exponentiated-Exponential Distribution for GPCI

Description

Built-in Exponentiated-Exponential Distribution for GPCI

Usage

dist_exponentiated_exponential()

Value

A gpci_dist object for Exponentiated Exponential distribution (Saha et al. 2022).

---

Built-in Gamma Distribution for GPCI

Description

Built-in Gamma Distribution for GPCI

Usage

dist_gamma()

Value

A gpci_dist object for Gamma distribution.

---

Built-in Logistic-Exponential Distribution for GPCI

Description

Built-in Logistic-Exponential Distribution for GPCI

Usage

dist_logistic_exponential()

Value

A gpci_dist object for Logistic-Exponential distribution (Lan & Leemis 2008, Alotaibi et al. 2022).

---

Built-in Normal Distribution for GPCI

Description

Built-in Normal Distribution for GPCI

Usage

dist_normal()

Value

A gpci_dist object for Normal distribution.

---

Built-in Weibull Distribution for GPCI

Description

Built-in Weibull Distribution for GPCI

Usage

dist_weibull()

Value

A gpci_dist object for Weibull distribution.

---

Fit Probability Distribution to Uncensored Data

Description

Fits a gpci_dist object to uncensored process data using Maximum Likelihood Estimation (MLE).

Usage

fit_distribution(data, distribution, method = "mle", init = NULL)

Arguments

data	Numeric vector of process observations.
distribution	A gpci_dist object.
method	Fitting method (default is "mle").
init	Optional initial parameter values list.

Value

An updated gpci_dist object containing estimated parameters.

Examples

data_x <- stats::rnorm(50, mean = 10, sd = 2)
dist_fitted <- fit_distribution(data_x, dist_normal())

---

High-Level User Interface Function for 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.

Usage

gpci_lindley(
  data,
  pdf = NULL,
  cdf = NULL,
  sf = NULL,
  prior = NULL,
  chain_length = 1000,
  burn_in = 200,
  thinning = 1,
  USL,
  LSL,
  target = (USL + LSL)/2,
  indices = NULL,
  B = 500
)

Arguments

data	Numeric vector of process observations.
pdf	Custom PDF function function(x, ...).
cdf	Custom CDF function function(x, ...).
sf	Custom Survival Function function(x, ...).
prior	Prior distribution hyperparameters list or log-prior function.
chain_length	Desired MCMC chain length (default 1000).
burn_in	MCMC burn-in iterations (default 200).
thinning	Thinning interval (default 1).
USL	Upper Specification Limit.
LSL	Lower Specification Limit.
target	Process target value.
indices	Character vector of GPCI names to evaluate.
B	Number of bootstrap replications (default 500).

Value

An object of class "gpciLindleyApprox".

Examples

gpci_lindley(
  data = stats::rnorm(50, 10, 1),
  pdf = function(x, mean = 0, sd = 1) stats::dnorm(x, mean, sd),
  cdf = function(x, mean = 0, sd = 1) stats::pnorm(x, mean, sd),
  chain_length = 300, burn_in = 50,
  USL = 13, LSL = 7, B = 100
)

---

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

x	Numeric vector representing an MCMC / simulation chain.
alpha	Significance level for the test (default is 0.05).
eps	Target maximum ratio of half-width to sample mean (default is 0.1).

Value

A list containing Cramér-von Mises statistic (stat), p-value (pvalue), stationarity test status (passed), half-width test status (hw_passed), half-width statistic (hw_stat), and convergence probability (convergence_prob).

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 / significance levels (e.g., 90%, 95%, 99%).

Usage

hpd_interval(x, prob = 0.95)

Arguments

x	Numeric vector of sample draws.
prob	Credibility level (1 - significance level). Default is 0.95.

Value

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

Examples

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

---

Lindley Approximation Method for Parameter and GPCI Bayesian Point Estimation

Description

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

Usage

lindley_approx(
  data,
  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

data	Numeric vector of uncensored process observations.
distribution	A gpci_dist object.
prior	Log-prior density function function(params) or a hyperparameter list. Default assumes non-informative flat or conjugate log-priors.
USL	Upper Specification Limit.
LSL	Lower Specification Limit.
target	Process target value. Default is (USL + LSL) / 2.
indices	Character vector of GPCI names to evaluate.
u	Parameter u for Vännman's Cp(u,v) index (default 1).
v	Parameter v for Vännman's Cp(u,v) index (default 1).
C0, C1, C2	Parameters for tolerance loss function.
P0	Baseline process yield (default 0.9973002).

Value

A list of class "gpciLindley" containing:

mle_params	Maximum Likelihood Estimates of parameters.
lindley_params	Lindley Bayes point estimates of parameters.
mle_gpci	GPCI point estimates computed at MLE.
lindley_gpci	Lindley Bayes point estimates of GPCIs.
cov_matrix	Estimated covariance matrix $\sigma_{ij}$.
hessian	Observed Hessian matrix.

Examples

data_x <- stats::rnorm(50, mean = 10, sd = 1)
dist_norm <- dist_normal()
fit_lindley <- lindley_approx(data = data_x, distribution = dist_norm, USL = 13, LSL = 7)

---

Lindley Approximation MCMC-Style Chain Generator for GPCIs

Description

Generates posterior parameter draws, GPCI chains, point estimates, HPD intervals, bootstrap confidence intervals, and convergence diagnostics for uncensored data using the Lindley approximation method combined with sampling and thinning.

Usage

lindley_gpci(
  data,
  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

data	Numeric vector of uncensored process data.
distribution	A gpci_dist object.
prior	Log-prior density function function(params) or hyperparameter list.
chain_length	Desired length of final retained chain after burn-in and thinning. Default is 1000.
burn_in	Number of initial burn-in samples to discard. Default is 200.
thinning	Thinning interval. Default is 1 (no thinning).
USL	Upper Specification Limit.
LSL	Lower Specification Limit.
target	Process target value (defaults to midpoint of USL and LSL).
indices	Character vector of GPCI names to evaluate. Default evaluates all available indices.
u, v	Parameters for Vännman's Cp(u,v) family (default 1).
C0, C1, C2	Parameters for loss function (defaults: 1, 0, 1).
P0	Baseline process yield (default 0.9973002).
B	Number of bootstrap replications for bootstrap CIs. Default is 500.

Value

An object of class "gpciLindleyApprox" containing:

param_chain	Matrix of retained parameter draws after burn-in and thinning.
gpci_chain	Data frame of retained GPCI chain values.
point_estimates	Point estimates of GPCIs under MLE.
lindley_estimates	Bayes point estimates of GPCIs under Lindley approximation.
bootstrap_results	Bootstrap confidence interval results.
specs	Specification limits and MCMC options.

Examples

data_x <- stats::rnorm(50, mean = 10, sd = 1)
dist_norm <- dist_normal()
fit_chain <- lindley_gpci(
  data = data_x,
  distribution = dist_norm,
  chain_length = 500,
  burn_in = 100,
  thinning = 1,
  USL = 13, LSL = 7, B = 200
)

---

Plot Visualizations for gpciLindleyApprox Objects

Description

Generates trace plots, posterior density distributions, and HPD / Bootstrap interval plots for Generalized Process Capability Indices (GPCIs) using ggplot2 or base R graphics.

Usage

## S3 method for class 'gpciLindleyApprox'
plot(x, index = NULL, type = c("trace", "density", "both"), ...)

Arguments

x	An object of class "gpciLindleyApprox".
index	Character string specifying which GPCI to plot. Default plots the first index.
type	Type of plot: "trace" for trace plot, "density" for posterior density, or "both".
...	Additional graphic parameters.

Value

A ggplot2 object or plot list.

Examples

data_x <- stats::rnorm(40, mean = 10, sd = 1)
fit_chain <- lindley_gpci(data = data_x, distribution = dist_normal(),
                         USL = 13, LSL = 7, chain_length = 200, burn_in = 50, B = 50)
plot(fit_chain, index = "Cpy")

---

Summary and Diagnostics for gpciLindleyApprox Objects

Description

Computes point estimates, Lindley Bayes estimates, Bias, MSE, Risk values (Linex + SEL), HPD intervals at 90%, 95%, and 99% confidence levels, Heidelberger and Welch MCMC convergence diagnostic, convergence probability, and Bootstrap confidence intervals.

Usage

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

Arguments

object	An object of class "gpciLindleyApprox".
...	Additional arguments.

Value

A data frame containing detailed diagnostic metrics for each evaluated GPCI.

Examples

data_x <- stats::rnorm(40, mean = 10, sd = 1)
fit_chain <- lindley_gpci(data = data_x, distribution = dist_normal(),
                         USL = 13, LSL = 7, chain_length = 300, burn_in = 50, B = 100)
summary(fit_chain)

---