Shrinkage Estimation in Multilevel Normal Models

Abstract

This review traces the evolution of theory that started when Charles Stein in 1955 [In Proc. 3rd Berkeley Sympos. Math. Statist. Probab. I (1956) 197–206, Univ. California Press] showed that using each separate sample mean from k ≥ 3 Normal populations to estimate its own population mean μ_i can be improved upon uniformly for every possible μ = (μ₁, …, μ_k)'. The dominating estimators, referred to here as being “Model-I minimax,” can be found by shrinking the sample means toward any constant vector. Admissible minimax shrinkage estimators were derived by Stein and others as posterior means based on a random effects model, “Model-II” here, wherein the μ_i values have their own distributions. Section 2 centers on Figure 2, which organizes a wide class of priors on the unknown Level-II hyperparameters that have been proved to yield admissible Model-I minimax shrinkage estimators in the “equal variance case.” Putting a flat prior on the Level-II variance is unique in this class for its scale-invariance and for its conjugacy, and it induces Stein’s harmonic prior (SHP) on μ_i.

Component estimators with real data, however, often have substantially “unequal variances.” While Model-I minimaxity is achievable in such cases, this standard requires estimators to have “reverse shrinkages,” as when the large variance component sample means shrink less (not more) than the more accurate ones. Section 3 explains how Model-II provides appropriate shrinkage patterns, and investigates especially estimators determined exactly or approximately from the posterior distributions based on the objective priors that produce Model-I minimaxity in the equal variances case. While correcting the reversed shrinkage defect, Model-II minimaxity can hold for every component. In a real example of hospital profiling data, the SHP prior is shown to provide estimators that are Model-II minimax, and posterior intervals that have adequate Model-II coverage, that is, both conditionally on every possible Level-II hyperparameter and for every individual component μ_i, i = 1, …, k.