Lawrence D. Brown was born on December 16, 1940 in Los Angeles, California. He obtained his Ph.D. in mathematics from Cornell University in 1964. He has been on the faculty of the University of California, Berkeley, Cornell University, Rutgers University and, most recently, the Wharton School of the University of Pennsylvania, where he holds the Miers Busch Professorship of Statistics. Professor Brown was President of the Institute of Mathematical Statistics in 1992–1993, Coeditor of The Annals of Statistics for 1995–1997 and gave the prestigious Wald Memorial Lectures in 1985. In 1990, Professor Brown was elected to the U.S. National Academy of Sciences. In 1993, Purdue University awarded him an honorary D.Sc. degree in recognition of his distinguished achievements, and in 2002 he was named winner of the Wilks Memorial Award of the American Statistical Association.

Professor Brown is probably best known for his extensive work on the admissibility of estimators of one or more parameters. He has also published research on a broad variety of other topics including general decision theory, sequential analysis, properties of exponential families, foundations of statistical inference, conditional confidence, interval estimation and Edgeworth expansions, and bioequivalence. His current interests include functional nonparametrics, analysis of census data and queuing theory as involved in the analysis of call-center data.

The article touches on two themes in which Larry Brown has been interested, namely foundations, and mathematical analysis of Bayesian decision theory.

In the section on foundations, a new formulation of the problem of testing is given, and is theoretically explored. The formulation strikes a balance between false discoveries and missed discoveries. Basic higher order asymptotic theory for the α level that this formulation would imply is then worked out. Accuracy is investigated and examples are given.

In the section on Bayesian decision theory, first, the Brown identities are connected to a set of inequalities of elliptic boundary value problems. It is shown by four specific results that sometimes a result in that field can lead to new results in Bayesian decision theory, and sometimes a result in decision theory can give surprisingly useful information about a problem in that field. For instance, Brown identities can provide amazingly good estimates of best constants in the Nash inequalities over Sobolev spaces.

The article ends with a result on the modern theory of high dimensional Gaussian mean estimation. By means of a triangulation of the Brown identity, the Rayleigh-Ritz variational formula of boundary value problems, and the famous Donsker-Varadhan result connecting the Rayleigh-Ritz formula to the absorption time of a Brownian motion into the boundary of a smooth bounded open domain, we show that the minimax risk of estimating the Gaussian mean can be approximated by chasing a Brownian motion to the boundary of the parameter space. This link should be tested by simulation.

The information inequality has been shown to be an effective tool for providing lower bounds for the minimax risk. Bounds based on the chi-square distance can sometimes offer a considerable improvement especially when applied iteratively. This paper compares these two methods in four examples including the bounded normal mean problem as well as obervations from a Poisson distribution.

The purpose of this research is to extend the results of Johnstone and MacGibbon [18, 19] to study the asymptotic behavior of the ratio of linear minimax risk to nonlinear minimax risk for the estimation of a Poisson mean that lies in a rescaled compact l¹-ellipsoid using the information-normalized quadratic loss function. This would be an analogue of Pinsker’s [24] result for l²-ellipsoids with quadratic loss in the Gaussian case. In view of the work of Brown et al. [7] which demonstrated the asymptotic equivalence of the problems of estimation under the same loss function of the intensity of a non-homogeneous Poisson process and estimation in the Gaussian white noise model with drift, some results concerning Poisson mean estimation with respect to quadratic loss are also included.

In this paper minimax lower bounds are derived for the estimation of the instantaneous volatility in three related high-frequency statistical models. These bounds are based on new upper bounds for the Kullback-Leibler divergence between two multivariate normal random variables along with a spectral analysis of the processes. A comparison with known upper bounds shows that these lower bounds are optimal. Our major finding is that the Gaussian microstructure noise introduces an additional degree of ill-posedness for each model, respectively.

We study a regression model with a huge number of interacting variables. We consider a specific approximation of the regression function under two assumptions: (i) there exists a sparse representation of the regression function in a suggested basis, (ii) there are no interactions outside of the set of the corresponding main effects. We suggest an hierarchical randomized search procedure for selection of variables and of their interactions. We show that given an initial estimator, an estimator with a similar prediction loss but with a smaller number of non-zero coordinates can be found.

Variable selection in high dimensional space has challenged many contemporary statistical problems from many frontiers of scientific disciplines. Recent technological advances have made it possible to collect a huge amount of covariate information such as microarray, proteomic and SNP data via bioimaging technology while observing survival information on patients in clinical studies. Thus, the same challenge applies in survival analysis in order to understand the association between genomics information and clinical information about the survival time. In this work, we extend the sure screening procedure [6] to Cox’s proportional hazards model with an iterative version available. Numerical simulation studies have shown encouraging performance of the proposed method in comparison with other techniques such as LASSO. This demonstrates the utility and versatility of the iterative sure independence screening scheme.

In Gaussian sequence models with Gaussian priors, we develop some simple examples to illustrate three perspectives on matching of posterior and frequentist probabilities when the dimension p increases with sample size n: (i) convergence of joint posterior distributions, (ii) behavior of a non-linear functional: squared error loss, and (iii) estimation of linear functionals. The three settings are progressively less demanding in terms of conditions needed for validity of the Bernstein-von Mises theorem.

Telephone call centers are data-rich environments that, until recently, have not received sustained attention from academics. For about a decade now, we have been fortunate to work with our colleague, mentor and friend, Larry Brown, on the collection and analysis of large call-center datasets. This work has provided many fascinating windows into the world of call-center operations, stimulating further research and affecting management practice. Larry’s inexhaustible curiosity and creativity, sharp insight and unique technical power, have continuously been an inspiration to us. We look forward to collaborating with and learning from him on many occasions to come.

In this paper, we study operational heterogeneity of call center agents. Our proxy for heterogeneity is agents’ service times (call durations), a performance measure that prevalently “enjoys" tight management control. Indeed, managers of large call centers argue that a 1-second increase/decrease in average service time can translate into additional/reduced operating costs on the order of millions of dollars per year.

We are motivated by an empirical analysis of call-center data, which identifies both short-term and long-term factors associated with agent heterogeneity. Operational consequences of such heterogeneity are then illustrated via discrete event simulation. This highlights the potential benefits of analyzing individual agents’ operational histories. We are thus naturally led to a detailed analysis of agents’ learning-curves, which reveals various learning patterns and opens up new research opportunities.

We discuss and review recent developments in the area of applied algebraic topology, such as persistent homology and barcodes. In particular, we discuss how these are related to understanding more about manifold learning from random point cloud data, the algebraic structure of simplicial complexes determined by random vertices and, in most detail, the algebraic topology of the excursion sets of random fields.

Multiple testing for pairwise comparisons in a one way fixed and balanced analysis of variance model is studied. Normality, independence and homogeneity of variance is assumed. Two sided alternatives are considered. The usual stepwise procedures are shown to lack an intuitive and important interval property for acceptance sections of individual tests. This renders them inadmissible in terms of both types of errors. Alternative procedures that do have the interval property are suggested. The new procedures are compared to the standard procedures in terms of power and in terms of practicality. One of the new procedures follows logical restrictions which may be a desirable property in some instances.

For the general Bayesian model uncertainty framework, the focus of this paper is on the development of model space priors which can compensate for redundancy between model classes, the so-called dilution priors proposed in George (1999). Several distinct approaches for dilution prior construction are suggested. One is based on tessellation determined neighborhoods, another on collinearity adjustments, and a third on pairwise distances between models.

Since the introduction of Akaike’s information criteria (AIC) in 1973, numerous information criteria have been developed and widely used in model selection. Many papers concerning the justification of various model selection criteria followed, particularly with respect to model selection error rates (the probability of selecting a wrong model). A model selection criterion is called consistent if the model selection error rate decreases to zero as the sample size increases to infinity. Otherwise, it is inconsistent. In this paper, we explore sufficient consistency conditions for information criteria in the nonparametric (logspline) and parametric model comparison setting, and discuss finite sample model selection error rates.

This article examines some classical optimality properties of various mid p–values. It is shown that the usual mid p–value arises naturally via a Rao–Blackwellization argument of a certain type of mid p–value. In 2 × 2 × K contingency tables and I × J tables with ordered alternatives it is shown that the type of alternative is crucial in the construction of an improved evidence assessment rule. It turns out that dimension of the sufficient statistic determines the amount of improvement gained by Rao–Blackwellization. In the 2 × 2 × K contingency tables and the uniform association model it is shown that the Cohen and Sackrowitz (1992) mid p–value can be improved. In each of these examples, the p–values are not based only on sufficient statistics. However, each can be shown to reduce to the usual mid p–value upon Rao–Blackwellization. Furthermore the mid p–value can, more generally, be derived as an optimal procedure for estimating the truth function using a UMP test. Hence the mid p–value serves a special role as a procedure suggested by both Neyman–Pearson and Rao–Blackwell theories as well as from a Bayesian perspective. Mid p–values for the matched pairs sample and logistic regression models are also studied.

Theory and methodology for nonparametric regression have been particularly well developed in the case of additive homoscedastic Gaussian noise. Inspired by asymptotic equivalence theory, there have been ongoing efforts in recent years to construct explicit procedures that turn other function estimation problems into a standard nonparametric regression with Gaussian noise. Then in principle any good Gaussian nonparametric regression method can be used to solve those more complicated nonparametric models. In particular, Brown, Cai and Zhou [3] considered nonparametric regression in natural exponential families with a quadratic variance function.

In this paper we extend the scope of Brown, Cai and Zhou [3] to general natural exponential families by introducing a new explicit procedure that is based on the variance stabilizing transformation. The new approach significantly reduces the bias of the inverse transformation and as a consequence it enables the method to be applicable to a wider class of exponential families. Combining this procedure with a wavelet block thresholding estimator for Gaussian nonparametric regression, we show that the resulting estimator enjoys a high degree of adaptivity and spatial adaptivity with near-optimal asymptotic performance over a broad range of Besov spaces.

We propose a method for incorporating variable selection into local polynomial regression. This can improve the accuracy of the regression by extending the bandwidth in directions corresponding to those variables judged to be unimportant. It also increases our understanding of the dataset by highlighting areas where these variables are redundant. The approach has the potential to effect complete variable removal as well as perform partial removal when a variable redundancy applies only to particular regions of the data. We define a nonparametric oracle property and show that this is more than satisfied by our approach under asymptotic analysis. The usefulness of the method is demonstrated through simulated and real data numerical examples.

Monte Carlo methods are widely used in numerical integration, and variance reduction plays a key role in Monte Carlo integration. This paper investigates variance reduction for Monte Carlo integration in both finite dimensional Euclidean space and infinite dimensional Wiener space. The proposed variance reduction approaches are to use basis functions to construct control variates for finite dimensional integrals and utilize Itô-Wiener chaos expansion to design antithetic variates and control variates for Wiener integrals. We establish the variances of the proposed Monte Carlo integration estimators and show that the proposed methods can achieve dramatic variance reduction in comparison with the basic Monte Carlo estimators. Examples are used to illustrate the performance of the proposed estimators.

We consider estimation of the mean vector, θ, of a spherically symmetric distribution with unknown scale parameter σ under scaled quadratic loss. We show minimaxity of generalized Bayes estimators corresponding to priors of the form π(‖θ‖²)η^b where η = 1 / σ², for π(⋅) superharmonic with a non decreasing Laplacian under conditions on b and weak moment conditions. Furthermore, these generalized Bayes estimators are independent of the underlying density and thus have the strong robustness property of being simultaneously generalized Bayes and minimax for the entire class of spherically symmetric distributions.

The performance of a number of empirical Bayes methods are examined for the in-season prediction of batting averages with the 2005 Major League baseball data. Among the methodologies considered are new general empirical Bayes estimators in homoscedastic and heteroscedastic partial linear models.