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

Abstract

Standard-Nutzungsbedingungen:Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden.Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen.Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Abstract Many different robust estimation approaches for the covariance or shape matrix of multivariate data have been established until today. Tyler's M-estimator has been recognized as the 'most robust' M-estimator for the shape matrix of elliptically symmetric distributed data. Tyler's Mestimators 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. Terms of use: Documents inKeywords: Covariance matrix; distribution-free estimation; missing data; robust estimation; shape matrix; sign-based estimator; Tyler's M-estimator. AMS Subject Classification: Primary 62H12, Secondary 62H20.1 Previous versions of the manuscript have been circulated under the title Distribution-free shape matrix estimation for incomplete data. AbstractMany different robust estimation approaches for the covariance or shape matrix of multivariate data have been established until today. Tyler's M-estimator has been recognized as the 'most robust' M-estimator for the shape matrix of elliptically symmetric distributed data.

Stochastic Period and Cohort Effect State-Space Mortality Models Incorporating Demographic Factors via Probabilistic Robust Principal Components

“…For the problem of only scatter estimation, (Tyler 1987) introduced a function u 2 (s) = d s , which is investigated in details by (Frahm and Jaekel 2010).…”

“…, y N does not contains values equal to µ 2. the empirical distribution measure F N of the sample satisfies F N (S) < Remark 8. Generally, to deal with lack of uniqueness of Equation (84), a common practice is to impose additional constraints such as |C| = 1 as in (Frahm and Jaekel 2010) or trC = 1 as in (Tyler 1987) or (Sun et al 2016). …”

Geodesic Convexity and Covariance Estimation

“…Proof: We have (18) where is just a definition of and is due to g-convexity of , and is based on the properties (19) for conforming matrices, and…”

“…This method has been successfully applied to different practical applications ranging from array processing to sensor networks. It has been generalized to other settings involving regularization [14]- [18] and incomplete data [19]. Additional recent contributions addressing covariance estimation in non Gaussian conditions include [20].…”

Robust Estimation of Multivariate Location and Scatter in the Presence of Missing Data

“…Unfortunately, as evidenced by our simulation studies (see Section 7), ERTBS is not consistent for normal data and remains sensitive to clusters of outliers. Frahma and Jaekel (2010) extended the location and scatter M-estimators proposed by Tyler (1987) for the case of partially missing observations. However, since the score function of these estimators is monotone, their complete data breakdown point is 1/(p + 1).…”