Improved Estimators for Coefficients in Linear Regression

Point estimators for the coefficients in orthogonal linear regression which are better than the ordinary least squares estimator are obtained when at least three coefficients are to be estimated. The measure of goodness of an estimator is the sum, or weighted sum, of the componentwise mean squared errors. Some of the new estimators have interpretations as estimators which depend upon preliminary tests of significance. These estimators may be especially appropriate when the independent variables fall into two sets or are ordered, as in polynomial regression or regression on principal components. The extension of the results to the general case of nonorthogonal regression is given; here the measure of goodness of an estimator is the mean of a quadratic form in the componentwise errors.