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Shrinkage Estimation of a Location Parameter for a Multivariate Skew Elliptic Distribution

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Abstract

The multivariate skew elliptic distributions include the multivariate skew-t distribution, which is represented as a mean- and scale-mixture distribution and is useful for analyzing skewed data with heavy tails. In the estimation of location parameters in the multivariate skew elliptic distributions, we derive minimax shrinkage estimators improving on the minimum risk location equivariant estimator relative to the quadratic loss function. Especially in the skew-t distribution, we suggest specific improved estimators where the conditions for their minimaxity do not depend on the degrees of freedom. We also study the case of a general elliptically symmetrical distribution when the covariance matrix is known up to an unknown multiple, but a residual vector is available to estimate the scale.

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Acknowledgments

We would like to thank the Editor, the Associate Editor and the two reviewers for valuable comments and helpful suggestions which led to an improved version of this paper. We would like to thank Mr. R. Yuasa for his help in numerical computations. This work was partially supported by a grant from the Simons Foundation (#418098 to William Strawderman). This work was partially supported by Grant-in-Aid for Scientific Research from Japan Society for the Promotion of Science (# 18K11188 and # 15H01943 to Tatsuya Kubokawa). Dominique Fourdrinier’s research is partially supported by the Russian National Foundation [#17-11-01049] and by the Tassili program [#18MDU105] between ENSSEA (Pôle Universitaire de Koléa, Algeria) and Université de Rouen (France).

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Authors and Affiliations

  1. UNIROUEN, UNIHAVRE INSA Rouen, Université de Normandie, LITIS; avenue de l’Université, BP 12, 76801, Saint-Étienne-du-Rouvray, France

    Dominique Fourdrinier

  2. University of Tokyo, Faculty of Economics, 7-3-1, Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan

    Tatsuya Kubokawa

  3. Rutgers University, Department of Statistics and Biostatistics, 501 Hill Center, Busch Campus, Piscataway, NJ, 08855, USA

    William E. Strawderman

Authors
  1. Dominique Fourdrinier
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  2. Tatsuya Kubokawa
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  3. William E. Strawderman
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Correspondence to Dominique Fourdrinier.

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Fourdrinier, D., Kubokawa, T. & Strawderman, W.E. Shrinkage Estimation of a Location Parameter for a Multivariate Skew Elliptic Distribution. Sankhya A 85, 808–828 (2023). https://doi.org/10.1007/s13171-022-00280-9

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  • DOI: https://doi.org/10.1007/s13171-022-00280-9

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