One-sample Hotelling's T-squared Test for Multivariate Mean

Given a multivariate sample \(X\) and hypothesized mean \(\mu_0\), it tests $$H_0 : \mu_x = \mu_0\quad vs\quad H_1 : \mu_x \neq \mu_0$$ using the procedure by Hotelling (1931).

mean1.1931Hotelling(X, mu0 = rep(0, ncol(X)))

Arguments

X

an \((n\times p)\) data matrix where each row is an observation.

mu0

a length-\(p\) mean vector of interest.

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

Hotelling H (1931). “The Generalization of Student's Ratio.” The Annals of Mathematical Statistics, 2(3), 360--378. ISSN 0003-4851.

Examples

## CRAN-purpose small example
smallX = matrix(rnorm(10*3),ncol=3)
mean1.1931Hotelling(smallX) # run the test
#> 
#> 	One-sample Hotelling's T-squared Test
#> 
#> data:  smallX
#> statistic = 4.1391, p-value = 0.42
#> alternative hypothesis: true mean is different from mu0.
#> 

if (FALSE) {
## empirical Type 1 error 
niter   = 1000
counter = rep(0,niter)  # record p-values
for (i in 1:niter){
  X = matrix(rnorm(50*5), ncol=5)
  counter[i] = ifelse(mean1.1931Hotelling(X)$p.value < 0.05, 1, 0)
}

## print the result
cat(paste("\n* Example for 'mean1.1931Hotelling'\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=""))
}