Mathematical Statistics: Basic Ideas and Selected Topics, Volume II presents important statistical concepts, methods, and tools not covered in the authors’ previous volume. This second volume focuses on inference in non- and semiparametric models. It not only reexamines the procedures introduced in the first volume from a more sophisticated point of view but also addresses new problems originating from the analysis of estimation of functions and other complex decision procedures and large-scale data analysis.
The book covers asymptotic efficiency in semiparametric models from the Le Cam and Fisherian points of view as well as some finite sample size optimality criteria based on Lehmann–Scheffé theory. It develops the theory of semiparametric maximum likelihood estimation with applications to areas such as survival analysis. It also discusses methods of inference based on sieve models and asymptotic testing theory. The remainder of the book is devoted to model and variable selection, Monte Carlo methods, nonparametric curve estimation, and prediction, classification, and machine learning topics. The necessary background material is included in an appendix.
Using the tools and methods developed in this textbook, students will be ready for advanced research in modern statistics. Numerous examples illustrate statistical modeling and inference concepts while end-of-chapter problems reinforce elementary concepts and introduce important new topics. As in Volume I, measure theory is not required for understanding.
The solutions to exercises for Volume II are included in the back of the book.
Check out Volume I for fundamental, classical statistical concepts leading to the material in this volume.
Table of Contents
INTRODUCTION AND EXAMPLES
Tests of Goodness of Fit and the Brownian Bridge
Testing Goodness of Fit to Parametric Hypotheses
Regular Parameters. Minimum Distance Estimates
Estimation of Irregular Parameters
Stein and Empirical Bayes Estimation
TOOLS FOR ASYMPTOTIC ANALYSIS
Weak Convergence in Function Spaces
The Delta Method in Infinite Dimensional Space
DISTRIBUTION-FREE, UNBIASED, AND EQUIVARIANT PROCEDURES
Similarity and Completeness
Invariance, Equivariance, and Minimax Procedures
INFERENCE IN SEMIPARAMETRIC MODELS
Estimation in Semiparametric Models
Asymptotics. Consistency, and Asymptotic Normality
Efficiency in Semiparametric Models
Tests and Empirical Process Theory
Asymptotic Properties of Likelihoods. Contiguity
MONTE CARLO METHODS
The Nature of Monte Carlo Methods
Three Basic Monte Carlo Methods
Markov Chain Monte Carlo
Applications of MCMC to Bayesian and Frequentist Inference
NONPARAMETRIC INFERENCE FOR FUNCTIONS OF ONE VARIABLE
Convolution Kernel Estimates on R
Minimum Contrast Estimates: Reducing Boundary Bias
Regularization and Nonlinear Density Estimates
Nonparametric Regression for One Covariate
PREDICTION AND MACHINE LEARNING
Classification and Prediction
Asymptotic Risk Criteria
Performance and Tuning via Cross Validation
Model Selection and Dimension Reduction
Topics Briefly Touched and Current Frontiers
APPENDIX D: SUPPLEMENTS TO TEXT
APPENDIX E: SOLUTIONS
Problems and Complements 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.
" . . . the authors have done a superb job of selecting topics comprising most of the essential knowledge needed formodern research. Furthermore, these modern topics are considered with greater depth and sophistication than is usual in a general purpose text. And throughout its pages the book does a good job of linking the mathematical developments to major examples. The choice of topics and examples, along with the depth of coverage are the most attractive features of this volume."
~RobertW. Keener, University of Michigan