ICSNP (version 1.1-2)

HR.Mest: Simultaneous Affine Equivariant Estimation of Multivariate Median and Tyler's Shape Matrix

Description

iterative algorithm that finds the affine equivariant multivariate median by estimating tyler.shape simultaneously.

Usage

HR.Mest(X, maxiter = 100, eps.scale = 1e-06, eps.center = 1e-06,
        na.action = na.fail)

Value

A list containing:

center

vector with the estimated loaction.

scatter

matrix of the estimated scatter.

Arguments

X

a numeric data frame or matrix.

maxiter

maximum number of iterations.

eps.scale

convergence tolerance for the Tyler's shape matrix subroutine.

eps.center

convergence tolerance for the location estimate.

na.action

a function which indicates what should happen when the data contain 'NA's. Default is to fail.

Author

Klaus Nordhausen and Seija Sirkia

Details

The algorithm follows the idea of Hettmansperger and Randles (2002). There are, however, some differences. This algorithm has the vector of marginal medians as starting point for the location and the starting shape matrix is Tyler's shape matrix based on the vector of marginal medians and has then a location step and a shape step which are:

location step k+1:

transforming the data as \(y=x V_{k}^{-\frac{1}{2}}\) and computing the spatial median \(\mu_y\) of y using the function spatial.median. Then retransforming \(\mu_y\) to the original scale \(\mu_{x,k+1}=\mu_y V_{k}^{\frac{1}{2}} \).

shape step k+1:

computing Tyler's shape matrix \(V_{k+1}\) with respect to \(\mu_{x,k+1}\) by using the function tyler.shape.

The algorithm stops when the difference between two subsequent location estimates is smaller than eps.center.

There is no proof that the algorithm converges.

References

Hettmansperger, T.P. and Randles, R.H. (2002), A practical affine equivariant multivariate median, Biometrika, 89, 851--860.

Examples

set.seed(654321)
cov.matrix <- matrix(c(3,2,1,2,4,-0.5,1,-0.5,2), ncol=3)
X <- rmvnorm(100, c(0,0,0), cov.matrix)
res <- HR.Mest(X)
colMeans(X)
res$center
cov.matrix/det(cov.matrix)^(1/3)
res$scatter
rm(.Random.seed)