On the Bayesianity of minimum risk equivariant estimator for location or scale parameters under a general convex and invariant loss function

Amir T Payandeh Najafabadi

doi:10.1080/23311835.2015.1023670

Amir T Payandeh Najafabadi

Open Access

https://doi.org/10.1080/23311835.2015.1023670

Copy DOI

Journal: Cogent Mathematics	Publication Date: Mar 17, 2015
Citations: 1	License type: cc-by

Affiliation: Shahid Beheshti University

Abstract

The Minimum Risk Equivariant (MRE), estimator is a widely used estimator which has several well-known theoretical and practical properties. It is well known that for the square error and absolute error loss functions, the MRE estimator is a generalized Bayes estimator. This article investigates the potential Bayesianity (or generalized Bayesianity) of the MRE estimator under a general convex and invariant loss function, ρ(·), for estimating the location and scale parameters of an unimodal density function.

Highlights

Compare with the uniform minimum variance unbiased estimator, the Minimum Risk Equivariant (MRE) estimator: (1) typically exists for convex loss function and even for non-restricted loss functions and (2) does not need to consider randomized estimators (Lehmann & Casella, 1998, p. 156)
The MRE estimator has a wide range of applications in Finite sampling framework (Chandrasekar & Sajesh, 2013; Ledoit & Wolf, 2013), Reliability (Chandrasekar & Sajesh, 2013; Wei, Song, Yan, & Mao, 2000), Regression and non-linear models (Grafarend, 2006; Hallin & Jurečková, 2012), Contingency tables (Lehmann & Casella, 1998), Economic Forecasting (Elliott & Timmermann, 2013), etc
This paper provides a class of prior distributions that their corresponding Bayes estimator under general convex and invariant loss function coincides with the MRE estimator for location or scale family of distributions

Summary

Introduction

Compare with the uniform minimum variance unbiased estimator, the Minimum Risk Equivariant (MRE) estimator: (1) typically exists for convex loss function and even for non-restricted loss functions and (2) does not need to consider randomized estimators (Lehmann & Casella, 1998, p. 156). This paper provides a class of prior distributions that their corresponding Bayes estimator under general convex and invariant loss function coincides with the MRE estimator for location or scale family of distributions. The following from Marchand and Payandeh (2011) recalls the Bayes estimator under generalized loss function for location parameter .

Results

Conclusion