Shrinkage estimation of location parameters in a multivariate skew-normal distribution

Abstract

This paper studies decision theoretic properties of Stein type shrinkage estimators in simultaneous estimation of location parameters in a multivariate skew-normal distribution with known skewness parameters under a quadratic loss. The benchmark estimator is the best location equivariant estimator which is minimax. A class of shrinkage estimators improving on the best location equivariant estimator is constructed when the dimension of the location parameters is larger than or equal to four. An empirical Bayes estimator is also derived, and motivated from the Bayesian procedure, we suggest a simple skew-adjusted shrinkage estimator and show its dominance property. The performances of these estimators are investigated by simulation.

KEYWORDS and phrases:

• Decision theory
• dominance result
• empirical Bayes
• mean mixture of normal distributions
• minimaxity
• multivariate skew-normal distribution
• quadratic loss function
• risk function

MSC 2010:

• Primary 62C15
• 62C20
• Secondary 62C12
• 62F10
• 62F15

Acknowledgments

We would like to thank the Editor, the Associate Editor and the reviewer for valuable comments and helpful suggestions which led to an improved version of this paper. This research was supported in part by Grant-in-Aid for Scientific Research from Japan Society for the Promotion of Science (# 18K11188, # 15H01943 and # 26330036 to Tatsuya Kubokawa). This work was partially supported by a grant from the Simons Foundation (#418098 to William Strawderman).