Two-sample Simultaneous Test of Mean and Variance by Pearson and Neyman (1930)

Given two univariate samples \(x\) and \(y\), it tests $$H_0 : \mu_x = \mu_y, \sigma_x^2 = \sigma_y^2 \quad vs \quad H_1 : \textrm{ not } H_0$$ by approximating the null distribution with Beta distribution using the first two moments matching.

mvar2.1930PN(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.

Examples

## CRAN-purpose small example
x = rnorm(10)
y = rnorm(10)
mvar2.1930PN(x, y)
#> 
#> 	Two-sample Simultaneous Test of Mean and Variance by Muirhead
#> 	Approximation (1982).
#> 
#> data:  x and y
#> statistic = 0.21062, p-value = 0.7396
#> 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(mvar2.1930PN(x,y)$p.value < 0.05, 1, 0)
}

## print the result
cat(paste("\n* Example for 'mvar2.1930PN'\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=""))
}