Two-sample Test for High-Dimensional Means by Dempster (1958, 1960)

Given two multivariate data \(X\) and \(Y\) of same dimension, it tests $$H_0 : \mu_x = \mu_y\quad vs\quad H_1 : \mu_x \neq \mu_y$$ using the procedure by Dempster (1958, 1960).

mean2.1958Dempster(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

Dempster AP (1958). “A High Dimensional Two Sample Significance Test.” The Annals of Mathematical Statistics, 29(4), 995--1010. ISSN 0003-4851.

Dempster AP (1960). “A Significance Test for the Separation of Two Highly Multivariate Small Samples.” Biometrics, 16(1), 41. ISSN 0006341X.

Examples

## CRAN-purpose small example
smallX = matrix(rnorm(10*3),ncol=3)
smallY = matrix(rnorm(10*3),ncol=3)
mean2.1958Dempster(smallX, smallY) # run the test
#> 
#> 	Two-sample Test for High-Dimensional Means by Dempster (1958, 1960).
#> 
#> data:  smallX and smallY
#> statistic = 3.9602, p-value = 3.744e-05
#> alternative hypothesis: two means are different.
#> 

# \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(mean2.1958Dempster(X,Y)$p.value < 0.05, 1, 0)
}

## print the result
cat(paste("\n* Example for 'mean2.1958Dempster'\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 'mean2.1958Dempster'
#> *
#> * number of rejections   : 88
#> * total number of trials : 1000
#> * empirical Type 1 error : 0.088
# }