MSE performance of the weighted average estimators consisting of shrinkage estimators when each individual regression coefficient is estimated

Abstract

In this paper, we analytically derive the exact formula for the mean squared error (MSE) of two weighted average (WA) estimators for each individual regression coefficient. Further, we execute numerical evaluations to investigate small sample properties of the WA estimators, and compare the MSE performance of the WA estimators with the other shrinkage estimators and the usual OLS estimator. Our numerical results show that (1) the WA estimators have smaller MSE than the other shrinkage estimators and the OLS estimator over a wide region of parameter space; (2) the range where the relative MSE of the WA estimator is smaller than that of the OLS estimator gets narrower as the number of explanatory variables k increases.

KeyWords:

• Mean squared error
• Stein-rule estimator
• minimum mean squared error estimator
• adjusted minimum mean squared error estimator
• weighted average estimator

Mathematics subject classification:

• 62J05
• 62J07

Acknowledgments

The authors thank the editor and the anonymous referee for their extensively valuable comments.

Additional information

Funding

This work was supported by the Natural Science Foundation of Fujian Province of China [2018J01114] and the Fundamental Research Funds for the Central Universities [Grant No. 20720171022] and the Key Projects of Philosophy and Social Sciences Research of Ministry of Education of China [Grant No. 16JZD016] and JSPS KAKENHI [Grant No. 26780136].