L1pack (version 0.52)

WH.Laplace: Wilson-Hilferty transformation

Description

Returns the Wilson-Hilferty transformation for multivariate Laplace deviates.

Usage

WH.Laplace(x, center, Scatter)

Arguments

x

object of class 'LaplaceFit' from which is extracted the estimated Mahalanobis distances of the fitted model. Also x can be a vector or matrix of data with, say, \(p\) columns.

center

mean vector of the distribution or data vector of length \(p\). Not required if x have class 'LaplaceFit'.

Scatter

Scatter matrix (\(p\) by \(p\)) of the distribution. Not required if x have class 'LaplaceFit'.

Details

Let \(T = D/(2p)\) be a Gamma distributed 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 $$z = \frac{T^{1/3} - (1 - \frac{1}{9p})}{(\frac{1}{9p})^{1/2}}% $$ and \(z\) is approximately distributed as a standard normal distribution. This is useful, for instance, in the construction of QQ-plots.

References

Osorio, F., Galea, M., Henriquez, C., Arellano-Valle, R. (2023). Addressing non-normality in multivariate analysis using the t-distribution. AStA Advances in Statistical Analysis 107, 785-813.

Terrell, G.R. (2003). The Wilson-Hilferty transformation is locally saddlepoint. Biometrika 90, 445-453.

Wilson, E.B., 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

Scatter <- matrix(c(1,.5,.5,1), ncol = 2)
Scatter

# generate the sample
y <- rmLaplace(n = 500, Scatter = Scatter)
fit <- LaplaceFit(y)
z <- WH.Laplace(fit)
par(pty = "s")
qqnorm(z, main = "Transformed distances QQ-plot")
abline(c(0,1), col = "red", lwd = 2)