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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 = (μ 1 ,...,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.

This article reviews the state of multiparameter shrinkage estimators with emphasis on the empirical Bayes viewpoint, particularly in the case of parametric prior distributions. Some successful applications of major importance are considered. Recent results concerning estimates of error and confidence intervals are described and illustrated with data.

Brad Efron's paper has inspired a return to the ideas behind Bayes, frequency and empirical Bayes. The latter preferably would not be limited to exchangeable models for the data and hyperparameters. Parallels are revealed between microarray analyses and profiling of hospitals, with advances suggesting more decision modeling for gene identification also. Then good multilevel and empirical Bayes models for random effects should be sought when regression toward the mean is anticipated.

Five of the six univariate natural exponential families (NEFs) with quadratic variance functions (QVFs), meaning that their variances are at most quadratic functions of their means, are the Normal, Poisson, Gamma, Binomial, and Negative Binomial distributions. The sixth is the NEF-CHS, the NEF generated from convolved Hyperbolic Secant distributions. These six NEF-QVFs and their relatives are unified in this article and in the main diagram via arrows that connect NEFs with many other named distributions. Relatives include all of Pearson's families of conjugate distributions (e.g., Inverted Gamma, Beta, F, and Skewed-t), conjugate mixtures (including two Polya urn schemes), and conditional distributions (including Hypergeometrics and Negative Hypergeometrics). Limit laws that also relate these distributions are indicated by solid arrows in Figure 1.

The Poisson model and analyses here feature nonexchangeable gamma distributions (although exchangeable following a scale transformation) for individual parameters, with standard deviations proportional to means. A relatively uninformative prior distribution for the shrinkage values eliminates the ill behavior of maximum likelihood estimators of the variance components. When tested in simulation studies, the resulting procedure provides better coverage probabilities and smaller risk than several other published rules, and thus works well from Bayesian and frequentist perspectives alike. The computations provide fast, accurate density approximations to individual parameters and to structural regression coefficients. The computer program is publicly available through Statlib.

This paper provides a new method and algorithm for making inferences about the parameters of a two-level multivariate normal hierarchical model. One has observed J p-dimensional vector outcomes, distributed at level 1 as multivariate normal with unknown mean vectors and with known covariance matrices. At level 2, the unknown mean vectors also have normal distributions, with common unknown covariance matrix A and with means depending on known covariates and on unknown regression coefficients. The algorithm samples independently from the marginal posterior distribution of A by using rejection procedures. Functions such as posterior means and covariances of the level 1 mean vectors and of the level 2 regression coefficient are estimated by averaging over posterior values calculated conditionally on each value of A drawn. This estimation accounts for the uncertainty in A, unlike standard restricted maximum likelihood empirical Bayes procedures. It is based on independent draws from the exact posterior distributions, unlike Gibbs sampling. The procedure is demonstrated for profiling hospitals based on patients' responses concerning p = 2 types of problems (non-surgical and surgical). The frequency operating characteristics of the rule corresponding to a particular vague multivariate prior distribution are shown via simulation to achieve their nominal values in that setting.

We have tested alternative models of the demand for medical care using experimental data. The estimated response of demand to insurance plan is sensitive to the model used. We therefore use a split-sample analysis and find that a model that more closely approximates distributional assumptions and uses a nonparametric retransformation factor performs better in terms of mean squared forecast error. Simpler models are inferior either because they are not robust to outliers (e.g., ANOVA, ANOCOVA), or because they are inconsistent when strong distributional assumptions are violated (e.g., a two-parameter Box-Cox transformation).

The notion of a conjugate family of distributions plays a very important role in the Bayesian approach to parametric inference. One of the main features of such a family is that it is closed under sampling, but a conjugate family often provides prior distributions which are tractable in various other respects. This paper is concerned with the properties of conjugate families for exponential family models. Special attention is given to the class of natural exponential families having a quadratic variance function, for which the theory is particularly fruitful. Several classes of conjugate families have been considered in the literature and here we describe some of their most interesting features. Relationships between such classes are also discussed. Our aim is to provide a unified approach to the theory of conjugate families for exponential family likelihoods. An important aspect of the theory concerns reparameterisations of the exponential family under consideration. We briefly review the concept of a conjugate parameterisation, which provides further insight into many of the properties discussed throughout the paper. Finally, further implications of these results for Bayesian conjugate analysis of exponential families are investigated.[PUBLICATION ABSTRACT]

The normal, Poisson, gamma, binomial, and negative binomial distributions are univariate natural exponential families with quadratic variance functions (the variance is at most a quadratic function of the mean). Only one other such family exists. Much theory is unified for these six natural exponential families by appeal to their quadratic variance property, including infinite divisibility, cumulants, orthogonal polynomials, large deviations, and limits in distribution.