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This article concerns the Bayes and frequentist aspects of empirical Bayes inference. Some of the ideas explored go back to Robbins in the 1950s, while others are current. Several examples are discussed, real and artificial, illustrating the two faces of empirical Bayes methodology: “oracle Bayes” shows empirical Bayes in its most frequentist mode, while “finite Bayes inference” is a fundamentally Bayesian application. In either case, modern theory and computation allow us to present a sharp finite-sample picture of what is at stake in an empirical Bayes analysis.
Efron’s elegant approach to $g$-modeling for empirical Bayes problems is contrasted with an implementation of the Kiefer–Wolfowitz nonparametric maximum likelihood estimator for mixture models for several examples. The latter approach has the advantage that it is free of tuning parameters and consequently provides a relatively simple complementary method.
This is a contribution to the discussion of the enlightening paper by Professor Efron. We focus on empirical Bayes interval estimation. We discuss the oracle interval estimation rules, the empirical Bayes estimation of the oracle rule and the computation. Some numerical results are reported.
We present some personal reflections on empirical Bayes/ compound decision (EB/CD) theory following Efron (2019). In particular, we consider the role of exchangeability in the EB/CD theory and how it can be achieved when there are covariates. We also discuss the interpretation of EB/CD confidence interval, the theoretical efficiency of the CD procedure, and the impact of sparsity assumptions.
This paper is focused on kernel regression when the response variable is the shape of a 3D object represented by a configuration matrix of landmarks. Regression methods on this shape space are not trivial because this space has a complex finite-dimensional Riemannian manifold structure (non-Euclidean). Papers about it are scarce in the literature, the majority of them are restricted to the case of a single explanatory variable, and many of them are based on the approximated tangent space. In this paper, there are several methodological innovations. The first one is the adaptation of the general method for kernel regression analysis in manifold-valued data to the three-dimensional case of Kendall’s shape space. The second one is its generalization to the multivariate case and the addressing of the curse-of-dimensionality problem. Finally, we propose bootstrap confidence intervals for prediction. A simulation study is carried out to check the goodness of the procedure, and a comparison with a current approach is performed. Then, it is applied to a 3D database obtained from an anthropometric survey of the Spanish child population with a potential application to online sales of children’s wear.
Zero-inflated nonnegative continuous (or semicontinuous) data arise frequently in biomedical, economical, and ecological studies. Examples include substance abuse, medical costs, medical care utilization, biomarkers (e.g., CD4 cell counts, coronary artery calcium scores), single cell gene expression rates, and (relative) abundance of microbiome. Such data are often characterized by the presence of a large portion of zero values and positive continuous values that are skewed to the right and heteroscedastic. Both of these features suggest that no simple parametric distribution may be suitable for modeling such type of outcomes. In this paper, we review statistical methods for analyzing zero-inflated nonnegative outcome data. We will start with the cross-sectional setting, discussing ways to separate zero and positive values and introducing flexible models to characterize right skewness and heteroscedasticity in the positive values. We will then present models of correlated zero-inflated nonnegative continuous data, using random effects to tackle the correlation on repeated measures from the same subject and that across different parts of the model. We will also discuss expansion to related topics, for example, zero-inflated count and survival data, nonlinear covariate effects, and joint models of longitudinal zero-inflated nonnegative continuous data and survival. Finally, we will present applications to three real datasets (i.e., microbiome, medical costs, and alcohol drinking) to illustrate these methods. Example code will be provided to facilitate applications of these methods.
In this paper, we retrace the recent history of statistics by analyzing all the papers published in five prestigious statistical journals since 1970, namely: The Annals of Statistics, Biometrika, Journal of the American Statistical Association, Journal of the Royal Statistical Society, Series B and Statistical Science. The aim is to construct a kind of “taxonomy” of the statistical papers by organizing and clustering them in main themes. In this sense being identified in a cluster means being important enough to be uncluttered in the vast and interconnected world of the statistical research. Since the main statistical research topics naturally born, evolve or die during time, we will also develop a dynamic clustering strategy, where a group in a time period is allowed to migrate or to merge into different groups in the following one. Results show that statistics is a very dynamic and evolving science, stimulated by the rise of new research questions and types of data.
In the United States, county-level estimates of crop yield, production, and acreage published by the United States Department of Agriculture’s National Agricultural Statistics Service (USDA NASS) play an important role in determining the value of payments allotted to farmers and ranchers enrolled in several federal programs. Given the importance of these official county-level crop estimates, NASS continually strives to improve its crops county estimates program in terms of accuracy, reliability and coverage. In 2015, NASS engaged a panel of experts convened under the auspices of the National Academies of Sciences, Engineering, and Medicine Committee on National Statistics (CNSTAT) for guidance on implementing models that may synthesize multiple sources of information into a single estimate, provide defensible measures of uncertainty, and potentially increase the number of publishable county estimates. The final report titled Improving Crop Estimates by Integrating Multiple Data Sources was released in 2017. This paper discusses several needs and requirements for NASS county-level crop estimates that were illuminated during the activities of the CNSTAT panel. A motivating example of planted acreage estimation in Illinois illustrates several challenges that NASS faces as it considers adopting any explicit model for official crops county estimates.
Instrumental variable analysis is a widely used method to estimate causal effects in the presence of unmeasured confounding. When the instruments, exposure and outcome are not measured in the same sample, Angrist and Krueger (J. Amer. Statist. Assoc.87 (1992) 328–336) suggested to use two-sample instrumental variable (TSIV) estimators that use sample moments from an instrument-exposure sample and an instrument-outcome sample. However, this method is biased if the two samples are from heterogeneous populations so that the distributions of the instruments are different. In linear structural equation models, we derive a new class of TSIV estimators that are robust to heterogeneous samples under the key assumption that the structural relations in the two samples are the same. The widely used two-sample two-stage least squares estimator belongs to this class. It is generally not asymptotically efficient, although we find that it performs similarly to the optimal TSIV estimator in most practical situations. We then attempt to relax the linearity assumption. We find that, unlike one-sample analyses, the TSIV estimator is not robust to misspecified exposure model. Additionally, to nonparametrically identify the magnitude of the causal effect, the noise in the exposure must have the same distributions in the two samples. However, this assumption is in general untestable because the exposure is not observed in one sample. Nonetheless, we may still identify the sign of the causal effect in the absence of homogeneity of the noise.
Rob Kass has been been on the faculty of the Department of Statistics at Carnegie Mellon since 1981; he joined the Center for the Neural Basis of Cognition (CNBC) in 1997, and the Machine Learning Department (in the School of Computer Science) in 2007. He served as Department Head of Statistics from 1995 to 2004 and served as Interim Co-Director of the CNBC 2015–2018. He became the Maurice Falk Professor of Statistics and Computational Neuroscience in 2016.
Kass has served as Chair of the Section for Bayesian Statistical Science of the American Statistical Association, Chair of the Statistics Section of the American Association for the Advancement of Science, founding Editor-in-Chief of the journal Bayesian Analysis and Executive Editor of Statistical Science. He is an elected Fellow of the American Statistical Association, the Institute of Mathematical Statistics and the American Association for the Advancement of Science. He has been recognized by the Institute for Scientific Information as one of the 10 most highly cited researchers, 1995–2005, in the category of mathematics. Kass is the recipient of the 2017 Fisher Award and lectureship by the Committee of the Presidents of the Statistical Societies. This interview took place at Carnegie Mellon University in November 2017.
Noel Cressie, FAA is Director of the Centre for Environmental Informatics in the National Institute for Applied Statistics Research Australia (NIASRA) and Distinguished Professor in the School of Mathematics and Applied Statistics at the University of Wollongong, Australia. He is also Adjunct Professor at the University of Missouri (USA), Affiliate of Org 398, Science Data Understanding, at NASA’s Jet Propulsion Laboratory (USA), and a member of the Science Team for NASA’s Orbiting Carbon Observatory-2 (OCO-2) satellite. Cressie was awarded a B.Sc. with First Class Honours in Mathematics in 1972 from the University of Western Australia, and an M.A. and Ph.D. in Statistics in 1973 and 1975, respectively, from Princeton University (USA). Two brief postdoctoral periods followed, at the Centre de Morphologie Mathématique, ENSMP, in Fontainebleau (France) from April 1975–September 1975, and at Imperial College, London (UK) from September 1975–January 1976. His past appointments have been at The Flinders University of South Australia from 1976–1983, at Iowa State University (USA) from 1983–1998, and at The Ohio State University (USA) from 1998–2012. He has authored or co-authored four books and more than 280 papers in peer-reviewed outlets, covering areas that include spatial and spatio-temporal statistics, environmental statistics, empirical-Bayesian and Bayesian methods including sequential design, goodness-of-fit, and remote sensing of the environment. Many of his papers also address important questions in the sciences. Cressie is a Fellow of the Australian Academy of Science, the American Statistical Association, the Institute of Mathematical Statistics, and the Spatial Econometrics Association, and he is an Elected Member of the International Statistical Institute. Noel Cressie’s refereed, unrefereed, and other publications are available at: https://niasra.uow.edu.au/cei/people/UOW232444.html.