Multivariate Huber's M-estimator and its symmetrized version

Description

Iterative algorithms to estimate M-estimators of location and scatter as well as symmetrized M-estimator using Huber's weight functions.

Usage

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mvhuberM(X, qg = 0.9, fixed.loc = FALSE, location = NULL, init =
NULL, steps = Inf, eps = 1e-06, maxiter = 100, na.action = na.fail)

symmhuber(X, qg = 0.9, init = NULL, steps = Inf, eps = 1e-6, 
maxiter = 100, na.action = na.fail)

Arguments

X

a matrix or a data frame

qg

a tuning parameter. The default is 0.9, see details

fixed.loc

a logical, see details

location

an optional vector giving the location of the data or the initial value for the location if it is estimated

init

an optional starting value for scatter

steps

fixed number of iteration steps to take, if Inf iteration is repeated until convergence (or until maxiter steps)

eps

tolerance for convergence

maxiter

maximum number of iteration steps. Ignored if steps is finite

na.action

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

Details

mvhuberM computes multivariate M-estimators of location and scatter using Huber's weight functions. The tuning parameter qg defines cutoff-point c for weight functions so that c=F^{-1}(q), where F is the cdf of chi^2-distribution with p degrees of freedom. The estimators with maximal breakdown point are obtained with the choice qg=F(p+1). If fixed.loc is set TRUE, scatter estimator is computed with fixed location given by location (default is column means).

symmhuber computes Huber's M-estimator of scatter using pairwise differences of the data therefore avoiding location estimation.

Value

mvhuberM returns a list with components

location

a vector

scatter

a matrix

symmhuber returns a matrix.

Author(s)

Klaus Nordhausen, klaus.nordhausen@uta.fi,
Jari Miettinen, jari.p.miettinen@jyu.fi

References

Huber, P.J. (1981), Robust Statistics, Wiley, New York.

Lopuhaa, H.P. (1989). On the relation between S-estimators and M-estimators of multivariate location and covariance. Annals of Statistics, 17, 1662-1683.

Sirkia, S., Taskinen, S., Oja, H. (2007) Symmetrised M-estimators of scatter. Journal of Multivariate Analysis, 98, 1611-1629.

Examples

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A<-matrix(c(1,2,-3,4,3,-2,-1,0,4),ncol=3)
X<-matrix(rnorm(1500),ncol=3)%*%t(A)
mvhuberM(X)
symmhuber(X)