Estimation of patterned covariance in the multivariate linear models: an outer product least-squares approach

Abstract

In this article, we present a framework of estimating patterned covariance of interest in the multivariate linear models. The main idea in it is to estimate a patterned covariance by minimizing a trace distance function between outer product of residuals and its expected value. The proposed framework can provide us explicit estimators, called outer product least-squares estimators, for parameters in the patterned covariance of the multivariate linear model without or with restrictions on regression coefficients. The outer product least-squares estimators enjoy the desired properties in finite and large samples, including unbiasedness, invariance, consistency and asymptotic normality. We still apply the framework to three special situations where their patterned covariances are the uniform correlation, a generalized uniform correlation and a general q-dependence structure, respectively. Simulation studies for three special cases illustrate that the proposed method is a competent alternative of the maximum likelihood method in finite size samples.

Keywords:

• estimation
• general q-dependence
• least squares
• multivariate linear model
• outer product least squares
• patterned covariance
• uniform correlation

AMS Classification:

• 62H12
• 62F12

Acknowledgements

The authors are deeply grateful to the anonymous referees and Associate Editor for their constructive comments which led to greatly improve this article. Liu's research was supported by grants of Shanghai University of Finance and Economics for excellent PhD (No. 2012950214). Hu's research was supported by National Natural Science Foundation of China Grants 10971126. The work is also partially supported by Program for Changjiang Scholars and Innovative Research Team in University (No. IRT13077), Shanghai University of Finance and Economics through Project 211 Phase IV and Shanghai Leading Academic Discipline Project (No. B803).