A generalization of Tyler's M-estimators to the case of incomplete data

Authors:
Gabriel Frahm;Uwe Jaekel
Affiliations:
University of Cologne, Department of Economic and Social Statistics, Albertus-Magnus-Platz, D-50923 Cologne, Germany;University of Applied Sciences Koblenz, RheinAhrCampus Remagen, Department of Mathematics and Technology, Südallee 2, D-53424 Remagen, Germany
Venue:
Computational Statistics & Data Analysis
Year:
2010

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Abstract

Many different robust estimation approaches for the covariance or shape matrix of multivariate data have been established. Tyler's M-estimator has been recognized as the 'most robust' M-estimator for the shape matrix of elliptically symmetric distributed data. Tyler's M-estimators for location and shape are generalized by taking account of incomplete data. It is shown that the shape matrix estimator remains distribution-free under the class of generalized elliptical distributions. Its asymptotic distribution is also derived and a fast algorithm, which works well even for high-dimensional data, is presented. A simulation study with clean and contaminated data covers the complete-data as well as the incomplete-data case, where the missing data are assumed to be MCAR, MAR, and NMAR.