The discovery of mean square error consistency of a ridge estimator

A ridge estimator has been known for its superiority over the least squares estimator. In classical asymptotic theory dealing with the number of variables p fixed and the sample size n-->[infinity], the ridge estimator is a biased estimator. Recently, high dimensional data, such as microarray, exhibits a very high dimension p and a much smaller sample size n. There are discussions about the behavior of the ridge estimator when both p and n tend to [infinity], but very few dealing with n fixed and p-->[infinity]. The latter situation seems more relevant to microarray data in practice. Here we outline and describe the asymptotic properties of the ridge estimator when the sample size n is fixed and the dimension p-->[infinity]. Under certain regularity conditions, mean square error (MSE) consistency of the ridge estimator is established. We also propose a variable screening method to eliminate variables which are unrelated to the outcome and prove the consistency of the screening procedure.