Examples and Problems in Mathematical Statistics E-Book

Shelemyahu Zacks

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Serie: Wiley Series in Probability and Statistics
• Sprache: Englisch

Seitenzahl: 875

Veröffentlichungsjahr: 2013

Beschreibung

Provides the necessary skills to solve problems in mathematical statistics through theory, concrete examples, and exercises

With a clear and detailed approach to the fundamentals of statistical theory, Examples and Problems in Mathematical Statistics uniquely bridges the gap between theory andapplication and presents numerous problem-solving examples that illustrate the relatednotations and proven results.

Written by an established authority in probability and mathematical statistics, each chapter begins with a theoretical presentation to introduce both the topic and the important results in an effort to aid in overall comprehension. Examples are then provided, followed by problems, and finally, solutions to some of the earlier problems. In addition, Examples and Problems in Mathematical Statistics features:

• Over 160 practical and interesting real-world examples from a variety of fields including engineering, mathematics, and statistics to help readers become proficient in theoretical problem solving
• More than 430 unique exercises with select solutions
• Key statistical inference topics, such as probability theory, statistical distributions, sufficient statistics, information in samples, testing statistical hypotheses, statistical estimation, confidence and tolerance intervals, large sample theory, and Bayesian analysis

Recommended for graduate-level courses in probability and statistical inference, Examples and Problems in Mathematical Statistics is also an ideal reference for applied statisticians and researchers.

Leseprobe

Contents

Cover

Series

Title Page

Copyright Page

Dedication

Preface

List of Random Variables

List of Abbreviations

Chapter 1: Basic Probability Theory

PART I: THEORY

1.1 OPERATIONS ON SETS

1.2 ALGEBRA AND σ–FIELDS

1.3 PROBABILITY SPACES

1.4 CONDITIONAL PROBABILITIES AND INDEPENDENCE

1.5 RANDOM VARIABLES AND THEIR DISTRIBUTIONS

1.6 THE LEBESGUE AND STIELTJES INTEGRALS

1.7 JOINT DISTRIBUTIONS, CONDITIONAL DISTRIBUTIONS AND INDEPENDENCE

1.8 MOMENTS AND RELATED FUNCTIONALS

1.9 MODES OF CONVERGENCE

1.10 WEAK CONVERGENCE

1.11 LAWS OF LARGE NUMBERS

1.12 CENTRAL LIMIT THEOREM

1.13 MISCELLANEOUS RESULTS

PART II: EXAMPLES

PART III: PROBLEMS

PART IV: SOLUTIONS TO SELECTED PROBLEMS

Chapter 2: Statistical Distributions

PART I: THEORY

2.1 INTRODUCTORY REMARKS

2.2 FAMILIES OF DISCRETE DISTRIBUTIONS

2.3 SOME FAMILIES OF CONTINUOUS DISTRIBUTIONS

2.4 TRANSFORMATIONS

2.5 VARIANCES AND COVARIANCES OF SAMPLE MOMENTS

2.6 DISCRETE MULTIVARIATE DISTRIBUTIONS

2.7 MULTINORMAL DISTRIBUTIONS

2.8 DISTRIBUTIONS OF SYMMETRIC QUADRATIC FORMS OF NORMAL VARIABLES

2.9 INDEPENDENCE OF LINEAR AND QUADRATIC FORMS OF NORMAL VARIABLES

2.10 THE ORDER STATISTICS

2.11 t–DISTRIBUTIONS

2.12 F–DISTRIBUTIONS

2.13 THE DISTRIBUTION OF THE SAMPLE CORRELATION

2.14 EXPONENTIAL TYPE FAMILIES

2.15 APPROXIMATING THE DISTRIBUTION OF THE SAMPLE MEAN: EDGEWORTH AND SADDLEPOINT APPROXIMATIONS

PART II: EXAMPLES

PART III: PROBLEMS

PART IV: SOLUTIONS TO SELECTED PROBLEMS

Chapter 3: Sufficient Statistics and the Information in Samples

PART I: THEORY

3.1 INTRODUCTION

3.2 DEFINITION AND CHARACTERIZATION OF SUFFICIENT STATISTICS

3.3 LIKELIHOOD FUNCTIONS AND MINIMAL SUFFICIENT STATISTICS

3.4 SUFFICIENT STATISTICS AND EXPONENTIAL TYPE FAMILIES

3.5 SUFFICIENCY AND COMPLETENESS

3.6 SUFFICIENCY AND ANCILLARITY

3.7 INFORMATION FUNCTIONS AND SUFFICIENCY

3.8 THE FISHER INFORMATION MATRIX

3.9 SENSITIVITY TO CHANGES IN PARAMETERS

PART II: EXAMPLES

PART III: PROBLEMS

PART IV: SOLUTIONS TO SELECTED PROBLEMS

Chapter 4: Testing Statistical Hypotheses

PART I: THEORY

4.1 THE GENERAL FRAMEWORK

4.2 THE NEYMAN–PEARSON FUNDAMENTAL LEMMA

4.3 TESTING ONE–SIDED COMPOSITE HYPOTHESES IN MLR MODELS

4.4 TESTING TWO–SIDED HYPOTHESES IN ONE–PARAMETER EXPONENTIAL FAMILIES

4.5 TESTING COMPOSITE HYPOTHESES WITH NUISANCE PARAMETERS—UNBIASED TESTS

4.6 LIKELIHOOD RATIO TESTS

4.7 THE ANALYSIS OF CONTINGENCY TABLES

4.8 SEQUENTIAL TESTING OF HYPOTHESES

PART II: EXAMPLES

PART III: PROBLEMS

PART IV: SOLUTIONS TO SELECTED PROBLEMS

Chapter 5: Statistical Estimation

PART I: THEORY

5.1 GENERAL DISCUSSION

5.2 UNBIASED ESTIMATORS

5.3 THE EFFICIENCY OF UNBIASED ESTIMATORS IN REGULAR CASES

5.4 BEST LINEAR UNBIASED AND LEAST–SQUARES ESTIMATORS

5.5 STABILIZING THE LSE: RIDGE REGRESSIONS

5.6 MAXIMUM LIKELIHOOD ESTIMATORS

5.7 EQUIVARIANT ESTIMATORS

5.8 ESTIMATING EQUATIONS

5.9 PRETEST ESTIMATORS

5.10 ROBUST ESTIMATION OF THE LOCATION AND SCALE PARAMETERS OF SYMMETRIC DISTRIBUTIONS

PART II: EXAMPLES

PART III: PROBLEMS

PART IV: SOLUTIONS OF SELECTED PROBLEMS

Chapter 6: Confidence and Tolerance Intervals

PART I: THEORY

6.1 GENERAL INTRODUCTION

6.2 THE CONSTRUCTION OF CONFIDENCE INTERVALS

6.3 OPTIMAL CONFIDENCE INTERVALS

6.4 TOLERANCE INTERVALS

6.5 DISTRIBUTION FREE CONFIDENCE AND TOLERANCE INTERVALS

6.6 SIMULTANEOUS CONFIDENCE INTERVALS

6.7 TWO–STAGE AND SEQUENTIAL SAMPLING FOR FIXED WIDTH CONFIDENCE INTERVALS

PART II: EXAMPLES

PART III: PROBLEMS

PART IV: SOLUTION TO SELECTED PROBLEMS

Chapter 7: Large Sample Theory for Estimation and Testing

PART I: THEORY

7.1 CONSISTENCY OF ESTIMATORS AND TESTS

7.2 CONSISTENCY OF THE MLE

7.3 ASYMPTOTIC NORMALITY AND EFFICIENCY OF CONSISTENT ESTIMATORS

7.4 SECOND–ORDER EFFICIENCY OF BAN ESTIMATORS

7.5 LARGE SAMPLE CONFIDENCE INTERVALS

7.6 EDGEWORTH AND SADDLEPOINT APPROXIMATIONS TO THE DISTRIBUTION OF THE MLE: ONE–PARAMETER CANONICAL EXPONENTIAL FAMILIES

7.7 LARGE SAMPLE TESTS

7.8 PITMAN’S ASYMPTOTIC EFFICIENCY OF TESTS

7.9 ASYMPTOTIC PROPERTIES OF SAMPLE QUANTILES

PART II: EXAMPLES

PART III: PROBLEMS

PART IV: SOLUTION OF SELECTED PROBLEMS

Chapter 8: Bayesian Analysis in Testing and Estimation

PART I: THEORY

8.1 THE BAYESIAN FRAMEWORK

8.2 BAYESIAN TESTING OF HYPOTHESIS

8.3 BAYESIAN CREDIBILITY AND PREDICTION INTERVALS

8.4 BAYESIAN ESTIMATION

8.5 APPROXIMATION METHODS

8.6 EMPIRICAL BAYES ESTIMATORS

PART II: EXAMPLES

PART III: PROBLEMS

PART IV: SOLUTIONS OF SELECTED PROBLEMS

Chapter 9: Advanced Topics in Estimation Theory

PART I: THEORY

9.1 MINIMAX ESTIMATORS

9.2 MINIMUM RISK EQUIVARIANT, BAYES EQUIVARIANT, AND STRUCTURAL ESTIMATORS

9.3 THE ADMISSIBILITY OF ESTIMATORS

PART II: EXAMPLES

PART III: PROBLEMS

PART IV: SOLUTIONS OF SELECTED PROBLEMS

Reference

Author Index

Subject Index

Wiley Series in Probability and Statistics

WILEY SERIES IN PROBABILITY AND STATISTICS

Established by WALTER A. SHEWHART and SAMUEL S. WILKS

Editor: David J. Balding, Noel A. C. Cressie, Garrett M. Fitzmaurice, Harvey Goldstein, Ian M. Johnstone, Geert Molenberghs, David W. Scott, Adrian F. M. Smith, Ruey S. Tsay, Sanford WeisbergEditors Emeriti: Vic Barnett, J. Staurt Hunter, Joseph B. Kadane, Jozef L. Teugels

A complete list of the titles in this series appears at the end of this volume.

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Library of Congress Cataloging-in-Publication Data:

Zacks, Shelemyahu, 1932- author.  Examples and problems in mathematical statistics / Shelemyahu Zacks.    pages cm  Summary: “This book presents examples that illustrate the theory of mathematical statistics and details how to apply the methods for solving problems” – Provided by publisher.  Includes bibliographical references and index.  ISBN 978-1-118-60550-9 (hardback)  1. Mathematical statistics–Problems, exercises, etc. I. Title.    QC32.Z265 2013    519.5–dc23

2013034492

ISBN: 9781118605509

To my wife Hanna,our sons Yuval and David,and their families, with love.

Preface

I have been teaching probability and mathematical statistics to graduate students for close to 50 years. In my career I realized that the most difficult task for students is solving problems. Bright students can generally grasp the theory easier than apply it. In order to overcome this hurdle, I used to write examples of solutions to problems and hand it to my students. I often wrote examples for the students based on my published research. Over the years I have accumulated a large number of such examples and problems. This book is aimed at sharing these examples and problems with the population of students, researchers, and teachers.

The book consists of nine chapters. Each chapter has four parts. The first part contains a short presentation of the theory. This is required especially for establishing notation and to provide a quick overview of the important results and references. The second part consists of examples. The examples follow the theoretical presentation. The third part consists of problems for solution, arranged by the corresponding sections of the theory part. The fourth part presents solutions to some selected problems. The solutions are generally not as detailed as the examples, but as such these are examples of solutions. I tried to demonstrate how to apply known results in order to solve problems elegantly. All together there are in the book 167 examples and 431 problems.

The emphasis in the book is on statistical inference. The first chapter on probability is especially important for students who have not had a course on advanced probability. Chapter Two is on the theory of distribution functions. This is basic to all developments in the book, and from my experience, it is important for all students to master this calculus of distributions. The chapter covers multivariate distributions, especially the multivariate normal; conditional distributions; techniques of determining variances and covariances of sample moments; the theory of exponential families; Edgeworth expansions and saddle–point approximations; and more. Chapter Three covers the theory of sufficient statistics, completeness of families of distributions, and the information in samples. In particular, it presents the Fisher information, the Kullback–Leibler information, and the Hellinger distance. Chapter Four provides a strong foundation in the theory of testing statistical hypotheses. The Wald SPRT is discussed there too. Chapter Five is focused on optimal point estimation of different kinds. Pitman estimators and equivariant estimators are also discussed. Chapter Six covers problems of efficient confidence intervals, in particular the problem of determining fixed–width confidence intervals by two–stage or sequential sampling. Chapter Seven covers techniques of large sample approximations, useful in estimation and testing. Chapter Eight is devoted to Bayesian analysis, including empirical Bayes theory. It highlights computational approximations by numerical analysis and simulations. Finally, Chapter Nine presents a few more advanced topics, such as minimaxity, admissibility, structural distributions, and the Stein–type estimators.

I would like to acknowledge with gratitude the contributions of my many ex–students, who toiled through these examples and problems and gave me their important feedback. In particular, I am very grateful and indebted to my colleagues, Professors A. Schick, Q. Yu, S. De, and A. Polunchenko, who carefully read parts of this book and provided important comments. Mrs. Marge Pratt skillfully typed several drafts of this book with patience and grace. To her I extend my heartfelt thanks. Finally, I would like to thank my wife Hanna for giving me the conditions and encouragement to do research and engage in scholarly writing.

SHELEMYAHU ZACKS

List of Random Variables

B(n,p)Binomial, with parameters n and pE(μ)Exponential with parameter μEV(λ, α)Extreme value with parameters λ and αF(v1, v2)Central F with parameters v1 and v2F(n1, n2;λ)Noncentral F with parameters v1, v2, λG(λ,p)Gamma with parameters λ and pH(M,N,n)Hyper–geometric with parameters M, N, nN(μ, V)Multinormal with mean vector μ and covariance matrix VN(μ,σ)Normal with mean μ and σNB(, v)Negative–binomial with parameters , and vP(λ)Poisson with parameter λR(a, b)Rectangular (uniform) with parameters a and bt[n;λ]Noncentral Student’s t with parameters n and λt[n;ξ,V]Multivariate t with parameters n, ξ and Vt[n]Student’s t with n degrees of freedomW(λ, α )Weibul with parameters λ and αβ (p,q)Beta with parameters p and qχ2[n,λ]Noncentral chi–squared with parameters n and λχ2[n]Chi–squared with n degrees of freedom

List of Abbreviations

a.s.Almost surelyANOVAAnalysis of variancec.d.f.Cumulative distribution functioncov(x,y)Covariance of X and YCIConfidence intervalCLTCentral limit theoremCPCoverage probabilityCRCramer Rao regularity conditionsE {X| Y}Conditional expected value of X, given YE {X}Expected value of XFIMFisher information matrixi.i.d.Independent identically distributedLBUELinear best unbiased estimateLCLLower confidence limitm.g.f.Moment generating functionm.s.s.Minimal sufficient statisticsMEEMoments equations estimatorMLEMaximum likelihood estimatorMLRMonotone likelihood ratioMPMost powerfulMSEMean squared errorMVUMinimum variance unbiasedOCOperating characteristicp.d.f.Probability density functionp.g.f.Probability generating functionP {E| A}Conditional probability of E, given AP {E}Probability of EPTEPre–test estimatorr.v.Random variableRHSRight–hand sides.v.Stopping variableSEStandard errorSLLNStrong law of large numbersSPRTSequential probability ratio testtr {A}trace of the matrix AUCLUpper control limitUMPUniformly most powerfulUMPIUniformly most powerful invariantUMPUUniformly most powerful unbiasedUMVUUniformly minimum variance unbiasedV {X| Y}Conditional variance of X, given YV {X}Variance of Xw.r.t.With respect toWLLNWeak law of large numbers