Conflict Between Pretest Estimators Induced by Three Large Sample Tests Under a Regression Model with Student t-error

We consider the preliminary test approach to the estimation of regression parameters, in a multiple-regression model with non-normal error. Here, we consider the multivariate t-distribution as the error distribution. The preliminary test estimators (PTEs) based on the Wald, likelihood ratio (LR) and Lagrangian multiplier (LM) tests are given. Their bias, mean-square error matrix M and weighted mean-square error (WMSE) functions are derived and compared. We show that there is conflict between the three estimators with respect to their bias, M and WMSE because of certain inequalities that exist inherently among the test statistics and because of approximating them by the same asymptotic distribution. In the neighbourhood of the null hypothesis the PTE based on the LM test has the smallest WMSE followed by the LR-based estimator, and the estimator based on the Wald test is the worst. However, the PTE based on the Ward test performs the best followed by the LR-based estimator when the parameter moves away from the subspace of the restriction and the LM-based estimator is the worst. It has been shown that in the choice of the smallest level of significance to yield the best estimator in the sense of the highest minimum guaranteed efficiency the Wald-based PTE dominates the other two estimators. Further, the optimum choice of the level of significance becomes the traditional choice by using the Ward test.