"Shrinkage" An Article for the Encyclopedia of Environmetrics

The problem of estimating the mean of a normal distribution is central to the practice of statistics. This problem is at the heart of many of the most common procedures used today, such as the analysis of variance or regression. If we have a random sample X1,..., Xn, from a normal popula-tion with mean 11 and variance a-2, the natural estimator of 11 is the sample mean X = (1/n) Ei Xi. A question of interest is whether this estimator is the best estimator of the parameter 11· When assessing the performance of an estimator, in particular whether it is best, it is necessary to have a criterion with which to measure it against. A most popular measure is squared error loss, where we measure the per-formance of an estimator d of a parameter (} by the function (1) L(B, d) = (B- d)2, which is called a loss function. Under the loss function (1), X has many optimality properties. For example, it is a minimax estimator of /1, meaning that of all estimators of /1, its loss has the smallest maximum value. There are other properties that X enjoys, including the property of admissibility. An estimator d of a parameter (} is an admissible estimator of(} under the loss L(B, d) if there is no other estimator d ' that satisfies Eo[L(B, d)] ~ Eo[L(B, d')] for all B, with strict inequality for some values of (}