MVTMLE function

M-estimator of Location and Scatter Using Weights Coming From the Multivariate t-distribution

The algorithm of this function is based on a partial Newton approach and should be faster than the traditional fixed-point algorithm. If the data follows a multivariate t-distribution with the correctly specified degrees of freedom this function gives the maximum likelihood estimate of location and scatter.

MVTMLE(X, nu = 1, location = TRUE, eps = 1e-06, maxiter = 100)

Arguments

• X: numeric data matrix or dataframe. Missing values are not allowed.
• nu: assumed degrees of freedom of the t-distribution. Default is '1' which corresponds to the Cauchy distribution.
• location: logical or numeric. If FALSE, it is assumed that the scatter should be computed wrt to the origin. If TRUE the location will be estimated and if it is a numeric vector it will be computed wrt to this vector.
• eps: convergence tolerance, which means that the algorithm stops when the Frobenius norm of the gradient is smaller than eps.
• maxiter: maximum number of iterations.

Details

The assumed degree of freedom nu must be at least 1 when the location and scatter should be estimated. If only the scatter is to be estimated, then it needs to be larger than zero only.

In case maxiter is reached before convergence, the estimate at that iteration is returned and a warning is given.

Returns

A list containing: - mu: Estimated location if location=TRUE, otherwise the user specified location.

• Sigma: Estimated scatter matrix.

• iter: Number of iterations of the algorithm.

References

Kent, J.T., Tyler, D.E. and Vardi, Y. (1994), A curious likelihoodidentity for the multivariate t-distribution, Communications inStatistics, Theory and Methods, 23, 441-453.

Duembgen, L., Nordhausen, K. and Schuhmacher, H. (2016), New algorithmsfor M-estimation of multivariate location and scatter, Journal ofMultivariate Analysis, 144, 200-217.\Sexpr[results=rd]{tools:::Rd_expr_doi("10.1016/j.jmva.2015.11.009")}

Author(s)

Lutz Duembgen and Klaus Nordhausen

See Also

cov.trob, tM, MVTMLEsymm

Examples

MVTMLE(longley)
# compare to
# library(ICS)
# tM(longley)
# library(MASS)
# cov.trob(longley, nu=1, tol = 1e-06, maxit = 100)