WH.Laplace {L1pack} | R Documentation |
Wilson-Hilferty transformation
Description
Returns the Wilson-Hilferty transformation for multivariate Laplace deviates.
Usage
WH.Laplace(x, center, Scatter)
Arguments
x |
object of class |
center |
mean vector of the distribution or data vector of length |
Scatter |
Scatter matrix ( |
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)