1st Edition

Mathematical Statistics Basic Ideas and Selected Topics, Volume II

By Peter .J. Bickel, Kjell A. Doksum Copyright 2016
    488 Pages 1 B/W Illustrations
    by Chapman & Hall

    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.

    INTRODUCTION AND EXAMPLES
    Tests of Goodness of Fit and the Brownian Bridge
    Testing Goodness of Fit to Parametric Hypotheses
    Regular Parameters. Minimum Distance Estimates
    Permutation Tests
    Estimation of Irregular Parameters
    Stein and Empirical Bayes Estimation
    Model Selection

    TOOLS FOR ASYMPTOTIC ANALYSIS
    Weak Convergence in Function Spaces
    The Delta Method in Infinite Dimensional Space
    Further Expansions

    DISTRIBUTION-FREE, UNBIASED, AND EQUIVARIANT PROCEDURES
    Introduction
    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
    The Bootstrap
    Markov Chain Monte Carlo
    Applications of MCMC to Bayesian and Frequentist Inference

    NONPARAMETRIC INFERENCE FOR FUNCTIONS OF ONE VARIABLE
    Introduction
    Convolution Kernel Estimates on R
    Minimum Contrast Estimates: Reducing Boundary Bias
    Regularization and Nonlinear Density Estimates
    Confidence Regions
    Nonparametric Regression for One Covariate

    PREDICTION AND MACHINE LEARNING
    Introduction
    Classification and Prediction
    Asymptotic Risk Criteria
    Oracle Inequalities
    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

    REFERENCES

    INDICES

    Problems and Complements appear at the end of each chapter.

    Biography

    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