#' Univariate Test of Normality by Shapiro and Francia (1972)
#'
#' Given an univariate sample \eqn{x}, it tests
#' \deqn{H_0 : x\textrm{ is from normal distribution} \quad vs\quad H_1 : \textrm{ not } H_0}
#' using a test procedure by Shapiro and Francia (1972), which is an approximation to Shapiro and Wilk (1965).
#'
#' @param x a length-\eqn{n} data vector.
#'
#' @return a (list) object of \code{S3} class \code{htest} containing: \describe{
#' \item{statistic}{a test statistic.}
#' \item{p.value}{\eqn{p}-value under \eqn{H_0}.}
#' \item{alternative}{alternative hypothesis.}
#' \item{method}{name of the test.}
#' \item{data.name}{name(s) of provided sample data.}
#' }
#'
#' @examples
#' ## CRAN-purpose small example
#' x = rnorm(10)
#' norm.1972SF(x) # run the test
#'
#' \donttest{
#' ## generate samples from several distributions
#' x = stats::runif(496) # uniform
#' y = stats::rgamma(496, shape=2) # gamma
#' z = stats::rlnorm(496) # log-normal
#'
#' ## test above samples
#' test.x = norm.1972SF(x) # uniform
#' test.y = norm.1972SF(y) # gamma
#' test.z = norm.1972SF(z) # log-normal
#' }
#'
#' @references
#' \insertRef{shapiro_approximate_1972}{SHT}
#'
#' @concept gof_normal
#' @export
norm.1972SF <- function(x){
##############################################################
# PREPROCESSING
check_1d(x) # univariate vector
DNAME <- deparse(substitute(x))
##############################################################
# COMPUTATION
x = sort(x) # in an increasing order
n = length(x)
n <- length(x)
if ((n < 5 || n > 5000)){
stop("* norm.1972SF : we only take care of sample size between (5,5000).")
}
y = qnorm(ppoints(n, a = 3/8))
W = cor(x, y)^2
u = log(n)
v = log(u)
mu = -1.2725 + 1.0521 * (v - u)
sig = 1.0308 - 0.26758 * (v + 2/u)
z = (log(1 - W) - mu)/sig
##############################################################
# REPORT
thestat = W
hname = "Univariate Test of Normality by Shapiro and Francia (1972)"
Ha = paste("Sample ", DNAME, " does not follow normal distribution.",sep="")
names(thestat) = "W"
pvalue = stats::pnorm(z, lower.tail = FALSE)
res = list(statistic=thestat, p.value=pvalue, alternative = Ha, method=hname, data.name = DNAME)
class(res) = "htest"
return(res)
}
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