Stein's Phenomenon

Stein established an amazing result contradicting the general belief that sample means should be optimal when estimating several population means simultaneously, even with normally distributed observations. This chapter describes an elegant and simple heuristic reasoning underlying Stein's phenomenon. The impact of Stein's phenomenon on statistical science for multidimensional problems is striking. Indeed, Stein's phenomenon has motivated numerous methodological and theoretical advances, some of which are taking place even now. This chapter highlights some such advances, mostly in the context of a normal probability model. An important methodological advance based on Stein's phenomenon is the creation of improved confidence sets. Many of the challenging statistical problems in modern science relate to the dimensionality of the data, specifically a large ensemble size accompanied by comparatively small sample sizes. An interesting illustration of Stein's phenomenon in modern science, which is considered in detail, arises in molecular systems biology. A major goal is to characterize the interplay among genes so that molecular mechanisms of diseases and associated cellular functions can be identified and described. Fulfillment of this goal requires estimating associations between pairs of genes and mapping the interplay among genes as a network.