Risk Comparison of Improved Estimators in a Linear Regression Model with MultivariatetErrors under Balanced Loss Function

Guikai Hu,Shenghua Yu,Qingguo Li

doi:10.1155/2014/129205

Abstract

Under a balanced loss function, we derive the explicit formulae of the risk of the Stein-rule (SR) estimator, the positive-part Stein-rule (PSR) estimator, the feasible minimum mean squared error (FMMSE) estimator, and the adjusted feasible minimum mean squared error (AFMMSE) estimator in a linear regression model with multivariateterrors. The results show that the PSR estimator dominates the SR estimator under the balanced loss and multivariateterrors. Also, our numerical results show that these estimators dominate the ordinary least squares (OLS) estimator when the weight of precision of estimation is larger than about half, and vice versa. Furthermore, the AFMMSE estimator dominates the PSR estimator in certain occasions.

Highlights

In the literature, many statisticians have studied the risk comparisons of various estimators in the linear model with normal errors and have generated substantial results
We can obtain the risk of SR, part Steinrule (PSR), feasible minimum mean squared error (FMMSE), and adjusted feasible minimum mean squared error (AFMMSE) estimators, respectively, and discuss their dominance properties
When the error term obeyed a multivariate normal distribution, Baranchik [4] proved that the PSR estimator dominated uniformly the SR estimator under the quadratic loss, and Ohtani [13] proved that the SRSV estimator dominated uniformly the SR estimator under a balanced loss function

Summary

Introduction

Many statisticians have studied the risk comparisons of various estimators in the linear model with normal errors and have generated substantial results. Under the mean squared error of prediction, Stein [2] and James and Stein [3] proved that the SR estimator dominates the OLS estimator when the numbers of explanatory variables are more than two and the MSE of the SR estimator is minimized if a = (k − 2)/(V + 2). We use this value of a hereafter. Much work has been done about the balanced loss risk comparisons of improved estimators in the normal linear model. We will give the explicit formulae for the balanced loss risk of these estimators and compare their sampling performance by theoretical and numerical analysis.

Balanced Loss Function and Risk

Numerical Analysis

Proof of Main Results