Two-sample Test for High-Dimensional Covariances by Li and Chen (2012)

Given two multivariate data \(X\) and \(Y\) of same dimension, it tests $$H_0 : \Sigma_x = \Sigma_y\quad vs\quad H_1 : \Sigma_x \neq \Sigma_y$$ using the procedure by Li and Chen (2012).

cov2.2012LC(X, Y, use.unbiased = TRUE)

Arguments

X

an \((n_x \times p)\) data matrix of 1st sample.

Y

an \((n_y \times p)\) data matrix of 2nd sample.

use.unbiased

a logical; TRUE to use up to 4th-order U-statistics as proposed in the paper, FALSE for faster run under an assumption that \(\mu_h = 0\) (default: TRUE).

Value

a (list) object of S3 class htest containing:

statistic

a test statistic.

p.value

\(p\)-value under \(H_0\).

alternative

alternative hypothesis.

method

name of the test.

data.name

name(s) of provided sample data.

References

Li J, Chen SX (2012). “Two sample tests for high-dimensional covariance matrices.” The Annals of Statistics, 40(2), 908--940. ISSN 0090-5364.

Examples

## CRAN-purpose small example
smallX = matrix(rnorm(10*4),ncol=5)
smallY = matrix(rnorm(10*4),ncol=5)
cov2.2012LC(smallX, smallY) # run the test
#> 
#> 	Two-sample Test for High-Dimensional Covariances by Li and Chen (2012)
#> 
#> data:  smallX and smallY
#> statistic = 2.4537, p-value = 0.00707
#> alternative hypothesis: two covariances are not equal.
#> 

if (FALSE) {
## empirical Type 1 error : use 'biased' version for faster computation
niter   = 1000
counter = rep(0,niter)
for (i in 1:niter){
  X = matrix(rnorm(500*25), ncol=10)
  Y = matrix(rnorm(500*25), ncol=10)
  
  counter[i] = ifelse(cov2.2012LC(X,Y,use.unbiased=FALSE)$p.value  < 0.05,1,0)
  print(paste0("iteration ",i,"/1000 complete.."))
}

## print the result
cat(paste("\n* Example for 'cov2.2012LC'\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=""))
}