Two-sample Simultaneous Test of Means and Covariances by Hyodo and Nishiyama (2018)

Given a multivariate sample \(X\), hypothesized mean \(\mu_0\) and covariance \(\Sigma_0\), it tests $$H_0 : \mu_x = \mu_y \textrm{ and } \Sigma_x = \Sigma_y \quad vs\quad H_1 : \textrm{ not } H_0$$ using the procedure by Hyodo and Nishiyama (2018) in a similar fashion to that of Liu et al. (2017) for one-sample test.

sim2.2018HN(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

Hyodo M, Nishiyama T (2018). “A simultaneous testing of the mean vector and the covariance matrix among two populations for high-dimensional data.” TEST, 27(3), 680--699. ISSN 1133-0686, 1863-8260.

Examples

## CRAN-purpose small example
smallX = matrix(rnorm(10*3),ncol=3)
smallY = matrix(rnorm(10*3),ncol=3)
sim2.2018HN(smallX, smallY) # run the test
#> 
#> 	Two-sample Simultaneous Test of Means and Covariances by Hyodo and
#> 	Nishiyama (2018)
#> 
#> data:  smallX and smallY
#> T = -1.2483, p-value = 0.8113
#> alternative hypothesis: both means and 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(121*10), ncol=10)
  Y = matrix(rnorm(169*10), ncol=10)
  counter[i] = ifelse(sim2.2018HN(X,Y)$p.value < 0.05, 1, 0)
}

## print the result
cat(paste("\n* Example for 'sim2.2018HN'\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 'sim2.2018HN'
#> *
#> * number of rejections   : 62
#> * total number of trials : 1000
#> * empirical Type 1 error : 0.062
# }