Two-sample Simultaneous Test of Mean and Variance by Perng and Littell (1976)

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 Fisher's method of merging two \(p\)-values.

mvar2.1976PL(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

Perng SK, Littell RC (1976). “A Test of Equality of Two Normal Population Means and Variances.” Journal of the American Statistical Association, 71(356), 968--971. ISSN 0162-1459, 1537-274X.

Examples

## CRAN-purpose small example
x = rnorm(10)
y = rnorm(10)
mvar2.1976PL(x, y)
#> 
#> 	Two-sample Simultaneous Test of Mean and Variance by Perng and Littell
#> 	(1976).
#> 
#> data:  x and y
#> statistic = 10.062, p-value = 0.0394
#> 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(mvar2.1976PL(x,y)$p.value < 0.05, 1, 0)
}

## print the result
cat(paste("\n* Example for 'mvar2.1976PL'\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=""))
}