fastmatrix (version 0.5-7721)

WH.normal: Wilson-Hilferty transformation for chi-squared variates

Description

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

Usage

WH.normal(x)

Arguments

x

vector or matrix of data with, say, \(p\) columns.

Details

Let \(T = D^2/p\) be a random variable, where \(D^2\) denotes the squared Mahalanobis distance defined as $$D^2 = (\bold{x} - \bold{\mu})^T \bold{\Sigma}^{-1} (\bold{x} - \bold{\mu})$$

Thus the Wilson-Hilferty transformation is given by $$z = \frac{T^{1/3} - (1 - \frac{2}{9p})}{(\frac{2}{9p})^{1/2}}$$ 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.

See Also

Mahalanobis

Examples

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