Addison-Wesley / Prentice Hall

Statistics

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Mathematical Statistics, Updated Printing, 2/E
Peter J. Bickel
Kjell A. Doksum

ISBN-10: 0132306379
ISBN-13: 9780132306379

Publisher: Prentice Hall
Copyright: 2007
Format: Cloth; 576 pp
Published: 04/24/2006

Suggested retail price: $133.40
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For graduate-level courses in Statistical Inference or Theoretical Statistics in departments of Statistics, Bio-Statistics, Economics, Computer Science, and Mathematics.


An updated printing! In response to feedback from faculty and students, some sections within the book have been rewritten. Also, a number of corrections have been made, further improving the accuracy of this outstanding textbook.


This updated classic, time-honored introduction to the theory and practice of statistics modeling and inference reflects the changing focus of contemporary Statistics. Coverage begins with the more general nonparametric point of view and then looks at parametric models as submodels of the nonparametric ones which can be described smoothly by Euclidean parameters. Although some computational issues are discussed, this is very much a book on theory. It relates theory to conceptual and technical issues encountered in practice, viewing theory as suggestive for practice, not prescriptive. It shows readers how assumptions which lead to neat theory may be unrealistic in practice.


KEY TOPICS: Statistical Models, Goals, and Performance Criteria. Methods of Estimation. Measures of Performance, Notions of Optimality, and Construction of Optimal Procedures in Simple Situations. Testing Statistical Hypotheses: Basic Theory. Asymptotic Approximations. Multiparameter Estimation, Testing and Confidence Regions. A Review of Basic Probability Theory. More Advanced Topics in Analysis and Probability. Matrix Algebra. 

  • NEW - More rigorous, yet accessible.

    • Adds the necessary rigor for sophisticated masters- or doctoral-level work, yet keeps in mind the mathematics background that today's students bring to the course.

  • NEW - Unified Viewpoint–Views all models, parametric, semi-parametric, and non-parametric from a “coordinate free” point of view.
  • NEW - More comprehensive coverage of key topics–E.g., multidimensional parameters, exponential families, algorithms (including EM), asymptotics and Bayesian methods.

    • Gives students a thorough understanding of statistical inference.

  • NEW - Computational issues discussed– Give a careful proof of the convergence of an algorithm (but the computer code is not supplied and there is no reference to standard statistical languages and packages).
  • NEW - 50% more problems–Problems gradually increase in level from routine to more challenging. Some problems cover important ideas and results not treated in the text.

    • Helps students to gain confidence in their problem solving abilities.

  • Comprehensive, self-contained treatment of the various aspects of statistics–E.g., modeling, frequentist and Bayesian analysis (developed side by side), optimality, prediction, and large sample theory and methods. Appendices provided needed results from probability and analysis.

    • Helps students develop a thorough understanding of many aspects of statistical inference, and provides instructors with a comprehensive, single-source teaching tool.

  • Large number and variety of problems–Ranges from routine to challenging. Provides many hints for the difficult problems.

    • Gives beginning-level students the opportunity to develop their statistical skills systematically, one level at a time, without becoming frustrated and overwhelmed.

  • Theory related to conceptual and technical issues encountered in practice–Views theory as suggestive for practice, not prescriptive.

    • Shows students how assumptions which lead to neat theory may be unrealistic in practice.

  • More rigorous, yet accessible.

    • Adds the necessary rigor for sophisticated masters- or doctoral-level work, yet keeps in mind the mathematics background that today's students bring to the course.

  • Unified Viewpoint–Views all models, parametric, semi-parametric, and non-parametric from a “coordinate free” point of view.
  • More comprehensive coverage of key topics–E.g., multidimensional parameters, exponential families, algorithms (including EM), asymptotics and Bayesian methods.

    • Gives students a thorough understanding of statistical inference.

  • Computational issues discussed– Give a careful proof of the convergence of an algorithm (but the computer code is not supplied and there is no reference to standard statistical languages and packages).
  • 50% more problems–Problems gradually increase in level from routine to more challenging. Some problems cover important ideas and results not treated in the text.

    • Helps students to gain confidence in their problem solving abilities.

(NOTE: Each chapter concludes with Problems and Complements, Notes, and References.)

1. Statistical Models, Goals, and Performance Criteria.

Data, Models, Parameters, and Statistics. Bayesian Models. The Decision Theoretic Framework. Prediction. Sufficiency. Exponential Families.


2. Methods of Estimation.

Basic Heuristics of Estimation. Minimum Contrast Estimates and Estimating Equations. Maximum Likelihood in Multiparameter Exponential Families. Algorithmic Issues.


3. Measures of Performance.

Introduction. Bayes Procedures. Minimax Procedures. Unbiased Estimation and Risk Inequalities. Nondecision Theoretic Criteria.


4. Testing and Confidence Regions.

Introduction. 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. Prediction Intervals. Likelihood Ratio Procedures.


5. Asymptotic Approximations.

Introduction: The Meaning and Uses of Asymptotics. Consistency. First- and Higher-Order Asymptotics: The Delta Method with Applications. Asymptotic Theory in One Dimension. Asymptotic Behavior and Optimality of the Posterior Distribution.


6. 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.

The Basic Model. Elementary Properties of Probability Models. Discrete Probability Models. Conditional Probability and Independence. Compound Experiments. Bernoulli and Multinomial Trials, Sampling with and without Replacement. Probabilities on Euclidean Space. Random Variables and Vectors: Transformations. Independence of Random Variables and Vectors. The Expectation of a Random Variable. Moments. Moment and Cumulant Generating Functions. Some Classical Discrete and Continuous Distributions. Modes of Convergence of Random Variables and Limit Theorems. Further Limit Theorems and Inequalities. Poisson Process.


Appendix B: Additional Topics in Probability and Analysis.

Conditioning by a Random Variable or Vector. Distribution Theory for Transformations of Random Vectors. Distribution Theory for Samples from a Normal Population. The Bivariate Normal Distribution. Moments of Random Vectors and Matrices. The Multivariate Normal Distribution. Convergence for Random Vectors: Op and Op Notation. Multivariate Calculus. Convexity and Inequalities. Topics in Matrix Theory and Elementary Hilbert Space Theory.


Appendix C: Tables.

The Standard Normal Distribution. Auxiliary Table of the Standard Normal Distribution. t Distribution Critical Values. X 2 Distribution Critical Values. F Distribution Critical Values.


Index.

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