Two-sample Test for Covariance Matrices by Cai, Liu, and Xia (2013)

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 Cai, Liu, and Xia (2013).

cov2.2013CLX(X, Y)

Arguments

X

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

Y

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

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

Cai T, Liu W, Xia Y (2013). “Two-Sample Covariance Matrix Testing and Support Recovery in High-Dimensional and Sparse Settings.” Journal of the American Statistical Association, 108(501), 265--277. ISSN 0162-1459, 1537-274X.

Examples

## CRAN-purpose small example
smallX = matrix(rnorm(10*3),ncol=3)
smallY = matrix(rnorm(10*3),ncol=3)
cov2.2013CLX(smallX, smallY) # run the test
#> 
#> 	Two-sample Test for Covariance Matrices by Cai, Liu, and Xia (2013)
#> 
#> data:  smallX and smallY
#> statistic = 1.2388, p-value = 0.6023
#> alternative hypothesis: two covariances are not equal.
#> 

# \donttest{
## empirical Type 1 error 
niter   = 1000
counter = rep(0,niter)  # record p-values
for (i in 1:niter){
  X = matrix(rnorm(50*5), ncol=10)
  Y = matrix(rnorm(50*5), ncol=10)
  
  counter[i] = ifelse(cov2.2013CLX(X, Y)$p.value < 0.05, 1, 0)
}

## print the result
cat(paste("\n* Example for 'cov2.2013CLX'\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=""))
#> 
#> * Example for 'cov2.2013CLX'
#> *
#> * number of rejections   : 65
#> * total number of trials : 1000
#> * empirical Type 1 error : 0.065
# }