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 pp fixed and the sample size n→∞n→∞, the ridge estimator is a biased estimator. Recently, high dimensional data, such as microarray, exhibits a very high dimension pp and a much smaller sample size nn. There are discussions about the behavior of the ridge estimator when both pp and nn tend to ∞∞, but very few dealing with nn fixed and p→∞p→∞. 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 nn is fixed and the dimension p→∞p→∞. 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.