A study of minimax shrinkage estimators dominating the James-Stein estimator under the balanced loss function

Abdelkader Benkhaled,Mekki Terbeche,Waleed Almutiry,Abdenour Hamdaoui,Mohammed Alshahrani

doi:10.1515/math-2022-0008

Abstract

Abstract One of the most common challenges in multivariate statistical analysis is estimating the mean parameters. A well-known approach of estimating the mean parameters is the maximum likelihood estimator (MLE). However, the MLE becomes inefficient in the case of having large-dimensional parameter space. A popular estimator that tackles this issue is the James-Stein estimator. Therefore, we aim to use the shrinkage method based on the balanced loss function to construct estimators for the mean parameters of the multivariate normal (MVN) distribution that dominates both the MLE and James-Stein estimators. Two classes of shrinkage estimators have been established that generalized the James-Stein estimator. We study their domination and minimaxity properties to the MLE and their performances to the James-Stein estimators. The efficiency of the proposed estimators is explored through simulation studies.

Highlights

Estimating the mean parameters is one of the most often encountered difficulties in multivariate statistical analysis
This paper introduces a new class of shrinkage estimators that dominate the James-Stein estimator and the maximum likelihood estimator (MLE)
We constructed a new class of shrinkage estimator that dominate the James-Stein estimator for the estimation of the mean θ of the multivariate normal (MVN) distribution Z ~ Nq(θ, σ2Iq), where σ2 is unknown

Summary

Introduction

Estimating the mean parameters is one of the most often encountered difficulties in multivariate statistical analysis. In the context of enhancing the mean of the MVN distribution, Khursheed [5] studied the domination and admissibility properties of the MLE of a family of shrinkage estimators. Several studies have examined the minimaxity and domination properties for various shrinkage estimators under the Bayesian framework, including Efron and Morris [8,9], Berger and Strawderman [10], Benkhaled and Hamdaoui [11], Hamdaoui et al [12,13], and Zinodiny et al [14] Most of these studies have used the quadratic loss function to compute the risk function. This paper introduces a new class of shrinkage estimators that dominate the James-Stein estimator and the MLE.

A class of minimax shrinkage estimators

The improved shrinkage estimators of the James-Stein estimator

Simulation results

1: Curves of the risk ratios

Conclusion