The mean squared error of the likelihood based estimating equations in the general linear model

Abstract

Mahmood and King [17] revealed that the marginal likelihood has the inherent property of unbiased estimating equations amongst a range of modified likelihood methods. In this paper we extend our investigation to deriving the mean squared error of the scores of the profile likelihood, the marginal likelihood, the conditional profile likelihood and the conditional profile restricted likelihood. In terms of minimum mean squared error, the estimating equation from the conditional profile restricted likelihood emerged as the preferred method. This provides further support to the implications of the findings of poor small-sample properties of Lagrange Multiplier (LM) tests in the literature which are based on biased estimating equations or having a larger mean squared error of the scores. We demonstrate that the relative error of the mean squared error between the conditional profile restricted and the marginal likelihood methods is negligible for increasingly larger samples. Amongst the unbiased estimating equations the minimum mean squared error criteria provides a clear choice of selecting the estimating equation for the purpose of estimation and testing.