Inadmissibility of Maximum Likelihood Estimators in Some Multiple Regression Problems with Three or More Independent Variables

Consider a multiple regression problem in which the dependent variable and (3 or more) independent variables have a joint normal distribution. This problem was investigated some time ago by Charles Stein, who proposed reasonable loss functions for various problems involving estimation of the regression coefficients and who obtained various minimax and admissibility results. In this paper we continue this investigation and establish the inadmissibility of the traditional maximum likelihood estimators. Inadmissibility is proved by exhibiting explicit procedures having lower risk than the corresponding maximum likelihood procedure. These results are given in Theorems 1 and 2 of Section 3.