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```
#This function implements Pseudo-Gaussian test when the location is specified.
PseudoGaussianLK = function(X, location) {
data_size = dim(X)
n = data_size[1] #sample size
d = data_size[2] #dimension
sigma = tyler_cov(X, location) #tyler estimator of the convariance matrix
sigma_root = spd_matrix_pow(sigma,-1 / 2) #the square root of the inverse of the covariance matrix
Z = sweep(X, 2, location) %*% sigma_root #data standardization
norms = apply(Z, 1, norm, type = "2") #norms of the standardized data
# The following lines calculate the test statistic as it is described in the article.
U = Z / norms
Z_sign_sq = array(0, dim = c(n, d))
Z_sign_sq = Z*Z*sign(Z)
ones = rep(1,n)
M4 = sum(norms^4)/n
const = d * (d + 2) / (3 * n * M4)
statistic = const * t(ones)%*%Z_sign_sq%*%t(Z_sign_sq)%*%ones #test statistic
p_val = 1 - stats::pchisq(statistic, df = d) #p value
output = list(statistic = statistic[[1]], p.value = p_val[[1]])
return(output)
}
#This function implements Pseudo-Gaussian test when the location is unspecified.
PseudoGaussianLU = function(X) {
data_size = dim(X)
n = data_size[1] #sample size
d = data_size[2] #dimension
theta = colMeans(X) #estimator of the mean
sigma = tyler_cov(X, theta) #tyler estimator of the convariance matrix
sigma_root = spd_matrix_pow(sigma, -1/2) #the square root of the inverse of the covariance matrix
Z = sweep(X, 2, theta)%*%sigma_root #data standardization
norms = apply(Z, 1, norm, type = "2") #norms of the standardized data
U = Z/norms # normalization of the standardized data
# The following lines calculate the test statistic as it is described in the article.
M1 = sum(norms)/n
M2 = sum(norms^2)/n
M3 = sum(norms^3)/n
M4 = sum(norms^4)/n
Si = Z*Z*sign(Z)
ck = 4 * gamma(d / 2) / ((d ^ 2 - 1) * sqrt(pi) * gamma((d - 1) / 2))
ones = rep(1, n)
centseq = ck * (d + 1) * M1 * t(ones)%*%Z - t(ones)%*%Si
centseq = centseq / sqrt(n)
Gam = 1/(3 * M4 / (d * (d + 2)) - 2 * ck ^ 2 * (d + 1) * M1 * M3 +
ck ^ 2 * (d + 1) ^ 2 / d * M1 ^ 2 * M2) * diag(d)
statistic = centseq %*% Gam %*% t(centseq) #test statistic
p_val = 1 - stats::pchisq(statistic, df = d) #p value
output = list(statistic = statistic[[1]], p.value = p_val[[1]])
}
#' Pseudo-Gaussian test for elliptical symmetry
#'
#' Tests for elliptical symmetry: specified and unspecified location.
#'
#' @param X A numeric matrix.
#' @param location A vector of location parameters.
#'
#' @return An object of class \code{"htest"} containing the following components:
#' \item{\code{statistic}}{The value of the test statistic.}
#' \item{\code{pvalue}}{The p-value of the test.}
#' \item{\code{alternative}}{A character string describing the alternative hypothesis.}
#' \item{\code{method}}{A character string indicating what type of test was performed.}
#'
#' @details
#' Note that \code{location} allows the user to specify the known location.
#' The default is set to \code{NA} which means that the unspecified location test will be performed unless the user specifies location.
#'
#' @section Background:
#' Pseudo-Gaussian tests for elliptical symmetry are based on Le Cam’s theory of statistical experiments.
#' They are most efficient against a multivariate form of Fechner-type asymmetry.
#' These tests require finite moments of order 4 and they have a simple asymptotic chi-squared distribution
#' under the null hypothesis of ellipticity.
#'
#'
#' @references
#' Cassart, D., Hallin, M. & Paindaveine, D., (2008). Optimal detection of Fechner-asymmetry. \emph{Journal of Statistical Planning and Inference}, \bold{138}, 2499-2525.
#'
#' Cassart, D., (2007). Optimal tests for symmetry. Ph.D. thesis, Univ. libre de Bruxelles, Brussels.
#'
#' @examples
#'
#' ## sepal width and length of the versicolor subset of the Iris data
#' X = datasets::iris[51:100,1:2]
#'
#' PseudoGaussian(X)
#'
#' @export
PseudoGaussian = function(X, location = NA) {
dname = deparse(substitute(X)) # get the data name
# The following condition cheks if data have the matrix form. If not, it tries to convert data into a matrix if possible.
if(!is.matrix(X)) {
X = as.matrix(X)
if (!(is.matrix(X) && length(X) > 1)){
stop("X is not in the valid matrix form.")
}
}
# The following condition checks if all matrix instances are numeric values.
else if(!is.numeric(X)){
stop('X has to take numeric values')
}
# The following condition checks if the location is specified.
if (any(is.na(location))) {
output = PseudoGaussianLU(X)
}
else{
output = PseudoGaussianLK(X, location)
}
#The following lines construct htest object 'res' which is the output of this function.
names(output$statistic) = 'statistic'
res <-
list(
method = 'Pseudo-Gaussian test for elliptical symmetry',
data.name = dname,
statistic = output$statistic,
p.value = output$p.value,
alternative = 'the distribution is not elliptically symmetric'
)
class(res) <- "htest"
return(res)
}
```

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