Improved estimation in a multivariate regression with measurement error

Abstract

In this paper, we study the estimation problem about the regression coefficients of a multivariate regression model with measurement errors under some uncertain restrictions. Specifically, we propose the unrestricted estimator (UE) and three restricted estimators (REs), and prove that they are all consistent for the true coefficients. We derive the asymptotic distributions of the proposed estimators under the sequence of local alternative restrictions. We also propose shrinkage estimators (SEs) to address the problem of the uncertainty of the restrictions. In addition, we establish the asymptotic distributional risk (ADR) of the proposed estimators and compare the risk performance of these estimators. It is established that the REs perform better than the UE only near the restriction, while they perform poorly as one moves farther away from the restriction. We also prove that SEs dominate the UE. These theoretical results are confirmed by simulations.

KEYWORDS:

• ADR
• asymptotic normality
• measurement error
• multivariate regression model
• restricted estimator
• shrinkage estimators
• Stein rules
• unrestricted estimator

Acknowledgements

The authors would like to thank the referees for helpful comments and useful insights. Further, Dr. Nkurunziza would like to acknowledge the financial support received from the Natural Sciences and Engineering Research Council of Canada (NSERC).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by Natural Sciences and Engineering Research Council of Canada [RGPIN-2019-04464].