Two-sample Simultaneous Test of Mean and Variance by Muirhead Approximation (1982)

Given two univariate samples \(x\) and \(y\), it tests $$H_0 : \mu_x = \mu_y, \sigma_x^2 = \sigma_y^2 \quad vs \quad H_1 : \textrm{ not } H_0$$ using Muirhead's approximation for small-sample problem.

mvar2.1982Muirhead(x, y)

Arguments

x

a length-\(n\) data vector.

y

a length-\(m\) data vector.

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

Muirhead RJ (1982). Aspects of multivariate statistical theory, Wiley series in probability and mathematical statistics. Wiley, New York. ISBN 978-0-471-09442-5.

Examples

## CRAN-purpose small example
x = rnorm(10)
y = rnorm(10)
mvar2.1982Muirhead(x, y)
#> 
#> 	Two-sample Simultaneous Test of Mean and Variance by Muirhead
#> 	Approximation (1982).
#> 
#> data:  x and y
#> statistic = 0.84364, p-value = 0.353
#> alternative hypothesis: true mean and variance of x are different from those of y.
#> 

if (FALSE) {
## empirical Type 1 error 
niter   = 1000
counter = rep(0,niter)  # record p-values
for (i in 1:niter){
  x = rnorm(100)  # sample x from N(0,1)
  y = rnorm(100)  # sample y from N(0,1)
  
  counter[i] = ifelse(mvar2.1982Muirhead(x,y)$p.value < 0.05, 1, 0)
}

## print the result
cat(paste("\n* Example for 'mvar2.1982Muirhead'\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=""))
}