WH.student | R Documentation |
Returns the Wilson-Hilferty transformation of random variables with F distribution.
WH.student(x, center, cov, eta = 0)
x |
object of class |
center |
mean vector of the distribution or second data vector of length p. Not required if |
cov |
covariance matrix (p by p) of the distribution. Not required if |
eta |
shape parameter of the multivariate t-distribution. By default the multivariate normal ( |
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.
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.
cov
, mahalanobis
, envelope
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)
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