Given univariate samples \(X_1~,\ldots,~X_k\), it tests $$H_0 : \Sigma_1 = \cdots \Sigma_k\quad vs\quad H_1 : \textrm{at least one equality does not hold}$$ using the procedure by Schott (2001) using Wald statistics. In the original paper, it provides 4 different test statistics for general elliptical distribution cases. However, we only deliver the first one with an assumption of multivariate normal population.
covk.2001Schott(dlist)a list of length \(k\) where each element is a sample matrix of same dimension.
a (list) object of S3 class htest containing:
a test statistic.
\(p\)-value under \(H_0\).
alternative hypothesis.
name of the test.
name(s) of provided sample data.
Schott JR (2001). “Some tests for the equality of covariance matrices.” Journal of Statistical Planning and Inference, 94(1), 25--36. ISSN 03783758.
## CRAN-purpose small example
tinylist = list()
for (i in 1:3){ # consider 3-sample case
tinylist[[i]] = matrix(rnorm(10*3),ncol=3)
}
covk.2001Schott(tinylist) # run the test
#>
#> Test for Homogeneity of Covariances by Schott (2001)
#>
#> data: tinylist
#> statistic = 14.806, p-value = 0.2522
#> alternative hypothesis: at least one of equalities does not hold.
#>
if (FALSE) {
## test when k=5 samples with (n,p) = (100,20)
## 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(100*20),ncol=20)
}
counter[i] = ifelse(covk.2001Schott(mylist)$p.value < 0.05, 1, 0)
}
## print the result
cat(paste("\n* Example for 'covk.2001Schott'\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=""))
}