SHT (version 0.1.9)

cov1.2012Fisher: One-sample Test for Covariance Matrix by Fisher (2012)

Description

Given a multivariate sample \(X\) and hypothesized covariance matrix \(\Sigma_0\), it tests $$H_0 : \Sigma_x = \Sigma_0\quad vs\quad H_1 : \Sigma_x \neq \Sigma_0$$ using the procedure by Fisher (2012). This method utilizes the generalized form of the inequality $$\frac{1}{p} \sum_{i=1}^p (\lambda_i^r - 1)^{2s} \ge 0$$ and offers two types of test statistics \(T_1\) and \(T_2\) corresponding to the case \((r,s)=(1,2)\) and \((2,1)\) respectively.

Usage

cov1.2012Fisher(X, Sigma0 = diag(ncol(X)), type)

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.

Arguments

X

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

Sigma0

a \((p\times p)\) given covariance matrix.

type

1 or 2 for corresponding statistic from the paper.

References

fisher_testing_2012SHT

Examples

## CRAN-purpose small example
smallX = matrix(rnorm(10*3),ncol=3)
cov1.2012Fisher(smallX) # run the test

# \donttest{
## empirical Type 1 error 
niter   = 1000
counter1 = rep(0,niter)  # p-values of the type 1
counter2 = rep(0,niter)  # p-values of the type 2
for (i in 1:niter){
  X = matrix(rnorm(50*5), ncol=50) # (n,p) = (5,50)
  counter1[i] = ifelse(cov1.2012Fisher(X, type=1)$p.value < 0.05, 1, 0)
  counter2[i] = ifelse(cov1.2012Fisher(X, type=2)$p.value < 0.05, 1, 0)
}

## print the result
cat(paste("\n* Example for 'cov1.2012Fisher' \n","*\n",
"* empirical error with statistic 1 : ", round(sum(counter1/niter),5),"\n",
"* empirical error with statistic 2 : ", round(sum(counter2/niter),5),"\n",sep=""))
# }