`wilson.hilferty`

Wilson-Hilferty transformation

Description

Returns the Wilson-Hilferty transformation of random variables with chi-squared distribution.

Usage

wilson.hilferty(x)

Arguments

Argument	Description
`x`	vector or matrix of data with, say, \(p\) columns.

Details

Let \(F = D^2/p\) be a random variable, where \(D^2\) denotes the squared Mahalanobis distance defined as

\[D^2 = (\mathbf{x} - \mathbf{\mu})^T \mathbf{\Sigma}^{-1} (\mathbf{x} - \mathbf{\mu})\]

Thus the Wilson-Hilferty transformation is given by

\[z = \frac{F^{1/3} - (1 - \frac{2}{9p})}{(\frac{2}{9p})^{1/2}}\]

and \(z\) is approximately distributed as a standard normal distribution. This is useful, for instance, in the construction of QQ-plots.

References

Wilson, E.B., and Hilferty, M.M. (1931). The distribution of chi-square. Proceedings of the National Academy of Sciences of the United States of America 17 , 684-688.

Examples

x <- iris[,1:4]
z <- wilson.hilferty(x)
par(pty = "s")
qqnorm(z, main = "Transformed distances Q-Q plot")
abline(c(0,1), col = "red", lwd = 2, lty = 2)

wilson.hilferty