IMS Collections 9: From Probability to Statistics and Back: High-Dimensional Models and Processes — A Festschrift in Honor of Jon A. Wellner

Editors: M. Banerjee, F. Bunea, J. Huang, V. Koltchinskii, M.H. Maathuis

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For more than thirty years, Jon A. Wellner has made outstanding contributions to several very active and important areas of statistics and probability. His results have been especially influential in semiparametric statistics, estimation and testing problems under shape constraints, empirical processes theory (both classical and abstract), survival analysis, biostatistics, bootstrap, probability in Banach spaces and high-dimensional probability. This Festschrift honors Jon Wellner on the occasion of his 65th birthday. Many of the papers included in this volume were presented at the conference “From Probability to Statistics and Back: High-Dimensional Models and Processes” that took place in Seattle, Washington in 2010. They cover a broad range of topics related, at various levels, to Jon’s work.

Jon’s contribution to the statistical arena is further underscored by his four highly influential (co-authored) books on empirical processes, semi-parametric models and nonparametric maximum likelihood estimation. The impact of his books on the discipline and the vital role that they played in communicating the power of empirical processes and semiparametric theory to the statistical community as effective tools for studying statistical models can hardly be exaggerated. Indeed, in this regard, he should be seen as one of the visionaries who helped unleash the potency of empirical process theory for solving hard theoretical problems in the statistical arena and which brought about a paradigm shift in the approach to a broad sphere of asymptotics. Jon has also been a prolific mentor with 27 graduated Ph.D. students (and one more being advised) at the time of going to press, many of whom have gone on to successful research careers at distinguished universities. In addition, he has been a mentor and source of inspiration to junior colleagues who were not his students and who, in many cases, are formidable names in the profession today.

NSF-CBMS Regional Conference Series in Probability and Statistics, Volume 9: Nonparametric Bayesian Inference

by Peter Müller and Abel Rodriguez

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These notes arose out of a short course at UC Santa Cruz in summer 2010. Like the course, the notes provide an overview of some popular Bayesian nonparametric (BNP) probability models. The discussion follows a logical development of many commonly used nonparametric Bayesian models as generalizations of the Dirichlet process (DP) in different directions, including Pólya tree (PT) models, species sampling models (SSM), dependent DP (DDP) models and product partition models (PPM). The selection of topics is subjective, simply driven by what the authors are familiar with. As a result, some useful and elegant classes of models such as normalized random measures with random increments (NRMIs) are reviewed only briefly.

We focus on BNP models for random probability measures, keeping for example a discussion of Gaussian process priors to only a brief review in the introductory chapter. Also, we put the emphasis on developing models, rather than a discussion of BNP data analysis for important statistical inference problems. However, some data analysis is introduced by way of short examples and in an introductory chapter. Inference for BNP models often requires computation-intensive implementations. Keeping the focus on models, we decided against a discussion of computational algorithms at much length. The only exception are posterior simulation schemes for Dirichlet process (DP) and DP mixture models. Finally, we do not discuss asymptotic results. These are important and non-trivial. Excellent recent reviews appear in the monograph by Ghosh and Ramamoorthi (2003) and a review paper by Ghoshal (2010).