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#' One-sample Simultaneous Likelihood Ratio Test of Mean and Variance
#'
#' Given two univariate samples \eqn{x} and \eqn{y}, it tests
#' \deqn{H_0 : \mu_x = \mu_0, \sigma_x^2 = \sigma_0^2 \quad vs \quad H_1 : \textrm{ not } H_0}
#' using likelihood ratio test.
#'
#' @param x a length-\eqn{n} data vector.
#' @param mu0 hypothesized mean \eqn{\mu_0}.
#' @param var0 hypothesized variance \eqn{\sigma_0^2}.
#'
#' @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
#' mvar1.LRT(rnorm(10))
#'
#' \dontrun{
#' ## empirical Type 1 error
#' niter = 1000
#' counter = rep(0,niter) # record p-values
#' for (i in 1:niter){
#' x = rnorm(100) # sample x from N(0,1)
#'
#' counter[i] = ifelse(mvar1.LRT(x)$p.value < 0.05, 1, 0)
#' }
#'
#' ## print the result
#' cat(paste("\n* Example for 'mvar1.LRT'\n","*\n",
#' "* number of rejections : ", sum(counter),"\n",
#' "* total number of trials : ", niter,"\n",
#' "* empirical Type 1 error : ",round(sum(counter/niter),5),"\n",sep=""))
#' }
#'
#' @concept mvar
#' @export
mvar1.LRT <- function(x, mu0=0, var0=1){
##############################################################
# Preprocessing & Parameters
DNAME = deparse(substitute(x))
check_1d(x) # univariate vector of 1st class
check_number(mu0) # check : univariate mean
check_number(var0) # check : univariate variance
if (var0<=0){
stop("* mvar1.LRT : var0 should be a nonnegative real number.")
}
##############################################################
# Computation
n = length(x)
xbar = base::mean(x)
s2 = sum((x-xbar)^2)/n
y1 = n*s2/var0
y2 = n*((xbar-mu0)^2)/var0
# statistic & p-value
loglbd = (n/2)*log(exp(1)/n) + (n/2)*log(y1) - ((y1+y2)/2)
thestat = -2*loglbd
pvalue = pchisq(thestat, df=2, lower.tail = FALSE)
##############################################################
# REPORT
hname = "One-sample Simultaneous Likelihood Ratio Test of Mean and Variance."
Ha = "true mean and variance of x are different from mu0 and var0."
names(thestat) = "statistic"
res = list(statistic=thestat, p.value=pvalue, alternative = Ha, method=hname, data.name = DNAME)
class(res) = "htest"
return(res)
}
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