Mathematical Statistics: Basic Ideas and Selected Topics, Volume I, Second Edition presents fundamental, classical statistical concepts at the doctorate level. It covers estimation, prediction, testing, confidence sets, Bayesian analysis, and the general approach of decision theory. This edition gives careful proofs of major results and explains how the theory sheds light on the properties of practical methods.
The book first discusses non- and semiparametric models before covering parameters and parametric models. It then offers a detailed treatment of maximum likelihood estimates (MLEs) and examines the theory of testing and confidence regions, including optimality theory for estimation and elementary robustness considerations. It next presents basic asymptotic approximations with one-dimensional parameter models as examples. The book also describes inference in multivariate (multiparameter) models, exploring asymptotic normality and optimality of MLEs, Wald and Rao statistics, generalized linear models, and more.
Mathematical Statistics: Basic Ideas and Selected Topics, Volume II will be published in 2015. It will present important statistical concepts, methods, and tools not covered in Volume I.
Table of Contents
STATISTICAL MODELS, GOALS, AND PERFORMANCE CRITERIA
Data, Models, Parameters, and Statistics
The Decision Theoretic Framework
METHODS OF ESTIMATION
Basic Heuristics of Estimation
Minimum Contrast Estimates and Estimating Equations
Maximum Likelihood in Multiparameter Exponential Families
MEASURES OF PERFORMANCE
Unbiased Estimation and Risk Inequalities
Nondecision Theoretic Criteria
TESTING AND CONFIDENCE REGIONS
Choosing a Test Statistic: The Neyman-Pearson Lemma
Uniformly Most Powerful Tests and Monotone Likelihood Ratio Models
Confidence Bounds, Intervals, and Regions
The Duality between Confidence Regions and Tests
Uniformly Most Accurate Confidence Bounds
Frequentist and Bayesian Formulations
Likelihood Ratio Procedures
Introduction: The Meaning and Uses of Asymptotics
First- and Higher-Order Asymptotics: The Delta Method with Applications
Asymptotic Theory in One Dimension
Asymptotic Behavior and Optimality of the Posterior Distribution
INFERENCE IN THE MULTIPARAMETER CASE
Inference for Gaussian Linear Models
Asymptotic Estimation Theory in p Dimensions
Large Sample Tests and Confidence Regions
Large Sample Methods for Discrete Data
Generalized Linear Models
Robustness Properties and Semiparametric Models
APPENDIX A: A REVIEW OF BASIC PROBABILITY THEORY
APPENDIX B: ADDITIONAL TOPICS IN PROBABILITY AND ANALYSIS
APPENDIX C: TABLES
Problems and Complements, Notes, and References appear at the end of each chapter.
Peter J. Bickel is a professor emeritus in the Department of Statistics and a professor in the Graduate School at the University of California, Berkeley. Dr. Bickel is a member of the American Academy of Arts and Sciences and the National Academy of Sciences. He has been a Guggenheim Fellow and MacArthur Fellow, a recipient of the COPSS Presidents’ Award, and president of the Bernoulli Society and the Institute of Mathematical Statistics. He holds honorary doctorate degrees from the Hebrew University of Jerusalem and ETH Zurich.
Kjell A. Doksum is a senior scientist in the Department of Statistics at the University of Wisconsin–Madison. His research encompasses the estimation of nonparametric regression and correlation curves, inference for global measures of association in semiparametric and nonparametric settings, the estimation of regression quantiles, statistical modeling and analysis of HIV data, the analysis of financial data, and Bayesian nonparametric inference.
"These methods are clearly explained by two outstanding statistical practitioners. … This book is well supported by the references, increasing its value as a guide through the often difficult world of mathematical statistics. …the authors consider key topics which include asymptotic efficiency in semiparametric models, semiparametric maximum likelihood estimation, proportional hazards regression models and Markov chain Monte Carlo methods."
— Receptos Pharmaceuticals, San Diego, 2016