Bayesian nonparametric regression with dependent wavelets has dual shrinkage properties: there is shrinkage through a dependent prior put on functional differences, and shrinkage through the setting of most of the wavelet coefficients to zero through Bayesian variable selection methods. The methodology can deal with unequally spaced data and is efficient because of the existence of fast moves in model space for the MCMC computation.

The methodology is illustrated on the problem of modeling the oscillations of Cepheid variable stars; these are a class of pulsating variable stars with the useful property that their periods of variability are strongly correlated with their absolute luminosity. Once this relationship has been calibrated, knowledge of the period gives knowledge of the luminosity. This makes these stars useful as “standard candles” for estimating distances in the universe.

This paper reviews advances in Stein-type shrinkage estimation for spherically symmetric distributions. Some emphasis is placed on developing intuition as to why shrinkage should work in location problems whether the underlying population is normal or not. Considerable attention is devoted to generalizing the “Stein lemma” which underlies much of the theoretical development of improved minimax estimation for spherically symmetric distributions. A main focus is on distributional robustness results in cases where a residual vector is available to estimate an unknown scale parameter, and, in particular, in finding estimators which are simultaneously generalized Bayes and minimax over large classes of spherically symmetric distributions. Some attention is also given to the problem of estimating a location vector restricted to lie in a polyhedral cone.

Shrinkage estimation has become a basic tool in the analysis of high-dimensional data. Historically and conceptually a key development toward this was the discovery of the inadmissibility of the usual estimator of a multivariate normal mean.

This article develops a geometrical explanation for this inadmissibility. By exploiting the spherical symmetry of the problem it is possible to effectively conceptualize the multidimensional setting in a two-dimensional framework that can be easily plotted and geometrically analyzed. We begin with the heuristic explanation for inadmissibility that was given by Stein [In Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, Vol. I (1956) 197–206, Univ. California Press]. Some geometric figures are included to make this reasoning more tangible. It is also explained why Stein’s argument falls short of yielding a proof of inadmissibility, even when the dimension, p, is much larger than p = 3.

We then extend the geometric idea to yield increasingly persuasive arguments for inadmissibility when p ≥ 3, albeit at the cost of increased geometric and computational detail.

Since Stein’s 1956 seminal paper, shrinkage has played a fundamental role in both parametric and nonparametric inference. This article discusses minimaxity and adaptive minimaxity in nonparametric function estimation. Three interrelated problems, function estimation under global integrated squared error, estimation under pointwise squared error, and nonparametric confidence intervals, are considered. Shrinkage is pivotal in the development of both the minimax theory and the adaptation theory.

While the three problems are closely connected and the minimax theories bear some similarities, the adaptation theories are strikingly different. For example, in a sharp contrast to adaptive point estimation, in many common settings there do not exist nonparametric confidence intervals that adapt to the unknown smoothness of the underlying function. A concise account of these theories is given. The connections as well as differences among these problems are discussed and illustrated through examples.

The possibility of improving on the usual multivariate normal confidence was first discussed in Stein (1962). Using the ideas of shrinkage, through Bayesian and empirical Bayesian arguments, domination results, both analytic and numerical, have been obtained. Here we trace some of the developments in confidence set estimation.

Let X be a random vector with distribution P_θ where θ is an unknown parameter. When estimating θ by some estimator φ(X) under a loss function L(θ, φ), classical decision theory advocates that such a decision rule should be used if it has suitable properties with respect to the frequentist risk R(θ, φ). However, after having observed X = x, instances arise in practice in which φ is to be accompanied by an assessment of its loss, L(θ, φ(x)), which is unobservable since θ is unknown. A common approach to this assessment is to consider estimation of L(θ, φ(x)) by an estimator δ, called a loss estimator. We present an expository development of loss estimation with substantial emphasis on the setting where the distributional context is normal and its extension to the case where the underlying distribution is spherically symmetric. Our overview covers improved loss estimators for least squares but primarily focuses on shrinkage estimators. Bayes estimation is also considered and comparisons are made with unbiased estimation.

In a remarkable series of papers beginning in 1956, Charles Stein set the stage for the future development of minimax shrinkage estimators of a multivariate normal mean under quadratic loss. More recently, parallel developments have seen the emergence of minimax shrinkage estimators of multivariate normal predictive densities under Kullback–Leibler risk. We here describe these parallels emphasizing the focus on Bayes procedures and the derivation of the superharmonic conditions for minimaxity as well as further developments of new minimax shrinkage predictive density estimators including multiple shrinkage estimators, empirical Bayes estimators, normal linear model regression estimators and nonparametric regression estimators.

The need for small area estimates is increasingly felt in both the public and private sectors in order to formulate their strategic plans. It is now widely recognized that direct small area survey estimates are highly unreliable owing to large standard errors and coefficients of variation. The reason behind this is that a survey is usually designed to achieve a specified level of accuracy at a higher level of geography than that of small areas. Lack of additional resources makes it almost imperative to use the same data to produce small area estimates. For example, if a survey is designed to estimate per capita income for a state, the same survey data need to be used to produce similar estimates for counties, subcounties and census divisions within that state. Thus, by necessity, small area estimation needs explicit, or at least implicit, use of models to link these areas. Improved small area estimates are found by “borrowing strength” from similar neighboring areas.

The key to small area estimation is shrinkage of direct estimates toward some regression estimates obtained by using in addition administrative records and other available sources of information. These shrinkage estimates can often be motivated from both a Bayesian and a frequentist point of view, and indeed in this particular context, it is possible to obtain at least an operational synthesis between the two paradigms. Thus, on one hand, while small area estimates can be developed using a hierarchical Bayesian or an empirical Bayesian approach, similar estimates are also found using the theory of best linear unbiased prediction (BLUP) or empirical best linear unbiased prediction (EBLUP).

The present article discusses primarily normal theory-based small area estimation techniques, and attempts a synthesis between both the Bayesian and the frequentist points of view. The results are mostly discussed for random effects models and their hierarchical Bayesian counterparts. A few miscellaneous remarks are made at the end describing the current research for more complex models including some nonnormal ones. Also provided are some pointers for future research.

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.

The Reverse Stein Effect is identified and illustrated: A statistician who shrinks his/her data toward a point chosen without reliable knowledge about the underlying value of the parameter to be estimated but based instead upon the observed data will not be protected by the minimax property of shrinkage estimators such as that of James and Stein, but instead will likely incur a greater error than if shrinkage were not used.

Donald (Don) Arthur Berry, born May 26, 1940 in Southbridge, Massachusetts, earned his A.B. degree in mathematics from Dartmouth College and his M.A. and Ph.D. in statistics from Yale University. He served first on the faculty at the University of Minnesota and subsequently held endowed chair positions at Duke University and The University of Texas M.D. Anderson Center. At the time of the interview he served as Head of the Division of Quantitative Sciences, and Chairman and Professor of the Department of Biostatistics at UT M.D. Anderson Center.

Don’s research deals with the theory and applications of statistics, especially Bayesian methods for sequential design of experiments. His work challenges the status quo, always striving to improve design and analysis of clinical trials, genetic modeling and the process of health-related decision making. His research impacts health research broadly, but has achieved the greatest influence in cancer research. As of 2010, he has published over 200 articles and 10 books and has mentored 24 Ph.D. and 16 M.S. students.

Don’s honors include fellowship election to the International Statistical Institute, the American Statistical Association and the Institute of Mathematical Statistics. He gave Presidential invited addresses to the Western North American Region of the International Biometric Society (New Mexico, 2004), the Canadian Statistical Society (Ottawa, 2006) and the Eastern North American Region of the International Biometric Society (Washington, 2008).

Don married Donna Berry in 1960. Together they raised six children, Don, Mike, Tim, Scott, Jennifer and Erin. Celebrating Don’s 70th birthday, the authors co-organized two invited sessions and a dinner reception at the ENAR 2010 in New Orleans. This interview occurred while his family, friends, colleagues and students gathered to celebrate his birthday and his contributions to statistics.