Given a multivariate sample \(X\) and hypothesized covariance matrix \(\Sigma_0\), it tests $$H_0 : \Sigma_x = \Sigma_0\quad vs\quad H_1 : \Sigma_x \neq \Sigma_0$$ using the procedure by Wu and Li (2015). They proposed to use \(m\) number of multiple random projections since only a single operation might attenuate the efficacy of the test.
an \((n\times p)\) data matrix where each row is an observation.
a \((p\times p)\) given covariance matrix.
the number of random projections to be applied.
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.
Wu T, Li P (2015). “Tests for High-Dimensional Covariance Matrices Using Random Matrix Projection.” arXiv:1511.01611 [stat].
## CRAN-purpose small example
smallX = matrix(rnorm(10*3),ncol=3)
cov1.2015WL(smallX) # run the test
#>
#> One-sample Test for Covariance Matrix by Wu and Li (2015).
#>
#> data: smallX
#> T1m = 0.32076, p-value = 1
#> alternative hypothesis: true covariance is different from Sigma0.
#>
# \donttest{
## empirical Type 1 error
## compare effects of m=5, 10, 50
niter = 1000
rec1 = rep(0,niter) # for m=5
rec2 = rep(0,niter) # m=10
rec3 = rep(0,niter) # m=50
for (i in 1:niter){
X = matrix(rnorm(50*10), ncol=50) # (n,p) = (10,50)
rec1[i] = ifelse(cov1.2015WL(X, m=5)$p.value < 0.05, 1, 0)
rec2[i] = ifelse(cov1.2015WL(X, m=10)$p.value < 0.05, 1, 0)
rec3[i] = ifelse(cov1.2015WL(X, m=50)$p.value < 0.05, 1, 0)
}
## print the result
cat(paste("\n* Example for 'cov1.2015WL'\n","*\n",
"* Type 1 error with m=5 : ",round(sum(rec1/niter),5),"\n",
"* Type 1 error with m=10 : ",round(sum(rec2/niter),5),"\n",
"* Type 1 error with m=50 : ",round(sum(rec3/niter),5),"\n",sep=""))
#>
#> * Example for 'cov1.2015WL'
#> *
#> * Type 1 error with m=5 : 0.04
#> * Type 1 error with m=10 : 0.042
#> * Type 1 error with m=50 : 0.046
# }