#' Test for Equality of Means by Schott (2007)
#'
#' Given univariate samples \eqn{X_1~,\ldots,~X_k}, it tests
#' \deqn{H_0 : \mu_1 = \cdots \mu_k\quad vs\quad H_1 : \textrm{at least one equality does not hold}}
#' using the procedure by Schott (2007). It can be considered as a generalization
#' of two-sample testing procedure proposed by \code{\link[SHT:mean2.1996BS]{Bai and Saranadasa (1996)}}.
#'
#' @param dlist a list of length \eqn{k} where each element is a sample matrix of same dimension.
#'
#' @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
#' tinylist = list()
#' for (i in 1:3){ # consider 3-sample case
#' tinylist[[i]] = matrix(rnorm(10*3),ncol=3)
#' }
#' meank.2007Schott(tinylist)
#'
#' \donttest{
#' ## test when k=5 samples with (n,p) = (10,50)
#' ## empirical Type 1 error
#' niter = 1000
#' counter = rep(0,niter) # record p-values
#' for (i in 1:niter){
#' mylist = list()
#' for (j in 1:5){
#' mylist[[j]] = matrix(rnorm(10*5),ncol=5)
#' }
#'
#' counter[i] = ifelse(meank.2007Schott(mylist)$p.value < 0.05, 1, 0)
#' }
#'
#' ## print the result
#' cat(paste("\n* Example for 'meank.2007Schott'\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=""))
#' }
#'
#'
#' @references
#' \insertRef{schott_highdimensional_2007}{SHT}
#'
#' @concept mean_multivariate
#' @export
meank.2007Schott <- function(dlist){
##############################################################
# PREPROCESSING AND PARAMETERS
check_dlistnd(dlist)
vec.ni = unlist(lapply(dlist, nrow)) # obs : per-class
n = sum(vec.ni) # obs : total number
p = ncol(dlist[[1]])
g = length(dlist)
##############################################################
# COMPUTATION : PRELIMINARY
vec.E = lapply(dlist, cov) # per-class covariance
E = array(0,c(p,p))
for (i in 1:g){
E = E + (vec.E[[i]]*vec.ni[i]) # sum of scatter
}
vec.Xbar = lapply(dlist, colMeans)
Xbar = rep(0,p)
for (i in 1:g){
Xbar = Xbar + (vec.ni[i]*vec.Xbar[[i]])
}
Xbar = Xbar/n
H = array(0,c(p,p))
for (i in 1:g){
bdf = as.vector(vec.Xbar[[i]]-Xbar) # bars' difference
H = H + outer(bdf,bdf)*vec.ni[i]
}
e = (n-g)
h = (g-1)
##############################################################
# COMPUTATION : MAIN PART
# 1. the statistic
Tnp = (1/sqrt(n-1))*((aux_trace(H)/h) - (aux_trace(E)/e))
# 2. variance term
a = (1/((e+2)*(e-1)))*(aux_trace(E%*%E) - (1/e)*(aux_trace(E)^2))
Tnp.var = (2/h)*(a/e)
# 3. adjusted statistic and p-value
thestat = Tnp
thestat.adj = Tnp/sqrt(Tnp.var)
pvalue = stats::pnorm(thestat.adj, lower.tail = FALSE)
##############################################################
# REPORT
hname = "Test for Equality of Means by Schott (2007)"
DNAME = deparse(substitute(dlist))
Ha = "one of equalities does not hold."
names(thestat) = "Tnp"
res = list(statistic=thestat, p.value=pvalue, alternative = Ha, method=hname, data.name = DNAME)
class(res) = "htest"
return(res)
}
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