wilson.hilferty: Wilson-Hilferty transformation

WH.studentR Documentation

Wilson-Hilferty transformation

Description

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

Usage

WH.student(x, center, cov, eta = 0)

Arguments

x

object of class 'studentFit' 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 second data vector of length p. Not required if x have class 'studentFit'.

cov

covariance matrix (p by p) of the distribution. Not required if x have class 'studentFit'.

eta

shape parameter of the multivariate t-distribution. By default the multivariate normal (eta = 0) is considered.

Details

Let F the following random variable:

F = D^2/(p(1-2η))

where D^2 denotes the squared Mahalanobis distance defined as

D^2 = (x - μ)^T Σ^-1 (x - μ)

Thus the Wilson-Hilferty transformation is given by

z = ((1 - 2η/9)F^1/3 - (1 - 2/(9p))) / (2η/9 F^2/3 + 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.

For eta = 0, we obtain

z = (F^1/3 - (1 - 2/(9p))) / (2/(9p))^1/2

which is the Wilson-Hilferty transformation for chi-square variables.

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. doi: 10.1007/s10182-022-00468-2

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

cov, mahalanobis, envelope

Examples

data(companies)
x <- companies
z <- WH.student(x, center = colMeans(x), cov = cov(x))
par(pty = "s")
qqnorm(z, main = "Transformed distances Q-Q plot")
abline(c(0,1), col = "red", lwd = 2)

MVT documentation built on Feb. 16, 2023, 8:29 p.m.

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