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#' @title One-Sample Cramér–von Mises Goodness-of-Fit Test
#'
#' @description
#' Performs the one-sample Cramér–von Mises goodness-of-fit (GoF) test to assess whether
#' a sample comes from a specified distribution using asymptotic p-value approximations.
#'
#' @param x A numeric vector of observations.
#' @param dist A character string specifying the theoretical distribution. Must be one of
#' \code{"norm"}, \code{"exp"}, \code{"unif"}, \code{"lnorm"}, \code{"weibull"},
#' \code{"gamma"}, \code{"t"}, or \code{"chisq"}.
#' @param ... Distribution parameters passed to the corresponding \code{p} functions
#' (e.g., \code{mean}, \code{sd}, \code{rate}, \code{df}, etc.).
#' @param eps A small value to truncate extreme p-values (default is \code{1e-15}).
#'
#' @return An object of class \code{"htest"} with the following components:
#' \describe{
#' \item{statistic}{The computed Cramér–von Mises test statistic.}
#' \item{p.value}{The asymptotic p-value.}
#' \item{method}{A description of the test and distribution.}
#' \item{data.name}{The name of the data vector.}
#' }
#'
#' @details
#' The test uses the Cramér–von Mises statistic to assess how well the empirical
#' distribution function (EDF) of the sample agrees with the cumulative distribution
#' function (CDF) of the specified theoretical distribution. The p-value is computed
#' using approximation formulas derived from the asymptotic distribution of the test statistic.
#'
#' @examples
#' set.seed(123)
#' x1 <- rnorm(500, mean = 0, sd = 1)
#' CVMgof2(x1, dist = "norm", mean = 0, sd = 1)
#'
#' x2 <- rexp(500, rate = 2)
#' CVMgof2(x2, dist = "exp", rate = 2)
#'
#' x3 <- runif(200, min = -1, max = 3)
#' CVMgof2(x3, dist = "unif", min = -1, max = 3)
#'
#' @export
CVMgof2 <- function(x,
dist = c("norm", "exp", "unif", "lnorm", "weibull", "gamma", "t", "chisq"),
...,
eps = 1e-15) {
dist <- match.arg(dist)
DNAME <- deparse(substitute(x))
x <- na.omit(x)
n <- length(x)
if (n < 1) stop("At least one observation is required.")
pfuns <- list(
norm = function(...) function(u) pnorm((u - list(...)[["mean"]]) / list(...)[["sd"]]),
exp = function(...) function(u) pexp(u, ...),
unif = function(...) function(u) punif(u, ...),
lnorm = function(...) function(u) plnorm(u, ...),
weibull = function(...) function(u) pweibull(u, ...),
gamma = function(...) function(u) pgamma(u, ...),
t = function(...) function(u) pt(u, ...),
chisq = function(...) function(u) pchisq(u, ...)
)
pfun <- pfuns[[dist]](...)
x_s <- sort(x)
p <- pfun(x_s)
p[p < eps] <- eps
p[p > 1 - eps] <- 1 - eps
i <- seq_len(n)
W <- (1 / (12 * n)) + sum((p - (2 * i - 1) / (2 * n))^2)
WW <- (1 + 0.5 / n) * W
if (WW < 0.0275) {
pval <- 1 - exp(-13.953 + 775.5 * WW - 12542.61 * WW^2)
} else if (WW < 0.051) {
pval <- 1 - exp(-5.903 + 179.546 * WW - 1515.29 * WW^2)
} else if (WW < 0.092) {
pval <- exp( 0.886 - 31.62 * WW + 10.897 * WW^2)
} else if (WW < 1.1) {
pval <- exp( 1.111 - 34.242 * WW + 12.832 * WW^2)
} else {
pval <- 7.37e-10
}
RVAL <- list(
statistic = c(W = WW),
p.value = pval,
method = paste("Cramer von Mises gof test for", dist, "distribution"),
data.name = DNAME
)
class(RVAL) <- "htest"
RVAL
}
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