Asymptotic efficiency of ridge estimator in linear and semiparametric linear models

Abstract

The linear model with a growing number of predictors arises in many contemporary scientific endeavor. In this article, we consider the commonly used ridge estimator in linear models. We propose analyzing the ridge estimator for a finite sample size n and a growing dimension p . The existence and asymptotic normality of the ridge estimator are established under some regularity conditions when p → ∞ . It also occurs that a strictly linear model is inadequate when some of the relations are believed to be of certain linear form while others are not easily parameterized, and thus a semiparametric partial linear model is considered. For these semiparametric partial linear models with p > n , we develop a procedure to estimate the linear coefficients as if the nonparametric part is not present. The asymptotic efficiency of the proposed estimator for the linear component is studied for p → ∞ . It is shown that the proposed estimator of the linear component asymptotically performs very well.