# wilson.hilferty: Wilson-Hilferty transformation In fastmatrix: Fast Computation of some Matrices Useful in Statistics

## Description

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

## Usage

 1  wilson.hilferty(x) 

## Arguments

 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 = (\bold{x} - \bold{μ})^T \bold{Σ}^{-1} (\bold{x} - \bold{μ})

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.

cov, Mahalanobis
 1 2 3 4 5 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)