ABSTRACT
The problem of estimating the regression coefficients in the usual multiple regression model is considered when it is apriori suspected that the coefficients may be restricted to a subspace. The preliminary test estimator (PTE) based on the Wald (W), Likelihood Ratio (LR), and Lagrangian Multiplier (LM) tests are given. Their bias, mean square error matrix (M), and risk function are derived and compared. In the neighbourhood of the null hypothesis the PTE based on the LM test has the smallest risk followed by the LR based estimator and the estimator based on the W test is the worst. However, the PTE based on the W 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. A table has been prepared for maximum and minimum guaranteed relative efficiency of the estimators corresponding to the three tests. This table allows one to determine optimum level of significance corresponding to the optimum estimator among the three. It has been shown that the optimum choice of the level of significance becomes the traditional choice by using the W test.
