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Introduction to Mathematical Statistics _7th.pdf

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    Cover
    Title Page
    Copyright Page
    Contents
    Preface
    1 Probability and Distributions
    1.1 Introduction
    1.2 Set Theory
    1.3 The Probability Set Function
    1.4 Conditional Probability and Independence
    1.5 Random Variables
    1.6 Discrete Random Variables
    1.6.1 Transformations
    1.7 Continuous RandomVariables
    1.7.1 Transformations
    1.8 Expectation of a Random Variable
    1.9 Some Special Expectations
    1.10 Important Inequalities
    2 Multivariate Distributions
    2.1 Distributions of Two Random Variables
    2.1.1 Expectation
    2.2 Transformations: Bivariate Random Variables
    2.3 Conditional Distributions and Expectations
    2.4 The Correlation Coefficient
    2.5 Independent Random Variables
    2.6 Extension to Several Random Variables
    2.6.1 *Multivariate Variance-Covariance Matrix
    2.7 Transformations for Several Random Variables
    2.8 Linear Combinations of Random Variables
    3 Some Special Distributions
    3.1 The Binomial and Related Distributions
    3.2 The Poisson Distribution
    3.3 The Γ, χ2, and β Distributions
    3.4 The Normal Distribution
    3.4.1 Contaminated Normals
    3.5 The Multivariate Normal Distribution
    3.5.1 *Applications
    3.6 t- and F-Distributions
    3.6.1 The t-distribution
    3.6.2 The F-distribution
    3.6.3 Student’s Theorem
    3.7 Mixture Distributions
    4 Some Elementary Statistical Inferences
    4.1 Sampling and Statistics
    4.1.1 HistogramEstimates of pmfs and pdfs
    4.2 Confidence Intervals
    4.2.1 Confidence Intervals for Difierence in Means
    4.2.2 Confidence Interval for Difference in Proportions
    4.3 Confidence Intervals for Parameters of Discrete Distributions
    4.4 Order Statistics
    4.4.1 Quantiles
    4.4.2 Confidence Intervals for Quantiles
    4.5 Introduction to Hypothesis Testing
    4.6 Additional Comments About Statistical Tests
    4.7 Chi-Square Tests
    4.8 The Method of Monte Carlo
    4.8.1 Accept–Reject Generation Algorithm
    4.9 Bootstrap Procedures
    4.9.1 Percentile Bootstrap Confidence Intervals
    4.9.2 Bootstrap Testing Procedures
    4.10 *Tolerance Limits for Distributions
    5 Consistency and Limiting Distributions
    5.1 Convergence in Probability
    5.2 Convergence in Distribution
    5.2.1 Bounded in Probability
    5.2.2 Δ-Method
    5.2.3 Moment Generating Function Technique
    5.3 Central Limit Theorem
    5.4 *Extensions to Multivariate Distributions
    6 Maximum Likelihood Methods
    6.1 Maximum Likelihood Estimation
    6.2 Rao–Cramér Lower Bound and Efficiency
    6.3 Maximum Likelihood Tests
    6.4 Multiparameter Case: Estimation
    6.5 Multiparameter Case: Testing
    6.6 The EM Algorithm
    7 Sufficiency
    7.1 Measures of Quality of Estimators
    7.2 A Sufficient Statistic for a Parameter
    7.3 Properties of a Sufficient Statistic
    7.4 Completeness and Uniqueness
    7.5 The Exponential Class of Distributions
    7.6 Functions of a Parameter
    7.7 The Case of Several Parameters
    7.8 Minimal Sufficiency and Ancillary Statistics
    7.9 Sufficiency, Completeness, and Independence
    8 Optimal Tests of Hypotheses
    8.1 Most Powerful Tests
    8.2 Uniformly Most Powerful Tests
    8.3 Likelihood Ratio Tests
    8.4 The Sequential Probability Ratio Test
    8.5 Minimax and Classification Procedures
    8.5.1 Minimax Procedures
    8.5.2 Classification
    9 Inferences About Normal Models
    9.1 Quadratic Forms
    9.2 One-Way ANOVA
    9.3 Noncentral X[sup(2)] and F-Distributions
    9.4 Multiple Comparisons
    9.5 The Analysis of Variance
    9.6 A Regression Problem
    9.7 A Test of Independence
    9.8 The Distributions of Certain Quadratic Forms
    9.9 The Independence of Certain Quadratic Forms
    10 Nonparametric and Robust Statistics
    10.1 Location Models
    10.2 Sample Median and the Sign Test
    10.2.1 Asymptotic Relative Efficiency
    10.2.2 Estimating Equations Based on the Sign Test
    10.2.3 Confidence Interval for the Median
    10.3 Signed-Rank Wilcoxon
    10.3.1 Asymptotic Relative Efficiency
    10.3.2 Estimating Equations Based on Signed-Rank Wilcoxon
    10.3.3 Confidence Interval for the Median
    10.4 Mann–Whitney–Wilcoxon Procedure
    10.4.1 Asymptotic Relative Efficiency
    10.4.2 Estimating Equations Based on the Mann–Whitney–Wilcoxon
    10.4.3 Confidence Interval for the Shift Parameter Δ
    10.5 General Rank Scores
    10.5.1 Efficacy
    10.5.2 Estimating Equations Based on General Scores
    10.5.3 Optimization: Best Estimates
    10.6 Adaptive Procedures
    10.7 Simple Linear Model
    10.8 Measures of Association
    10.8.1 Kendall’s T
    10.8.2 Spearman’s Rho
    10.9 Robust Concepts
    10.9.1 Location Model
    10.9.2 Linear Model
    11 Bayesian Statistics
    11.1 Subjective Probability
    11.2 Bayesian Procedures
    11.2.1 Prior and Posterior Distributions
    11.2.2 Bayesian Point Estimation
    11.2.3 Bayesian Interval Estimation
    11.2.4 Bayesian Testing Procedures
    11.2.5 Bayesian Sequential Procedures
    11.3 More Bayesian Terminology and Ideas
    11.4 Gibbs Sampler
    11.5 Modern Bayesian Methods
    11.5.1 Empirical Bayes
    A: Mathematical Comments
    A.1 Regularity Conditions
    A.2 Sequences
    B: R Functions
    C: Tables of Distributions
    D: Lists of Common Distributions
    E: References
    F: Answers to Selected Exercises
    Index
    A
    B
    C
    D
    E
    F
    G
    H
    I
    J
    K
    L
    M
    N
    O
    P
    Q
    R
    S
    T
    U
    V
    W
    Z
    Introduction to Mathematical Statistics Seventh Edition Robert V. Hogg University of Iowa Joseph W. McKean Western Michigan University Allen T. Craig Late Professor of Statistics University of Iowa Boston Columbus Indianapolis New York San Francisco Upper Saddle River Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montreal Toronto Delhi Mexico City Sao Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo
    Editor in Chief: Deirdre Lynch Acquisitions Editor: Christopher Cummings Sponsoring Editor: Christina Lepre Editorial Assistant: Sonia Ashraf Senior Managing Editor: Karen Wernholm Senior Production Project Manager: Beth Houston Digital Assets Manager: Marianne Groth Marketing Manager: Erin K. Lane Marketing Coordinator: Kathleen DeChavez Senior Author Support/Technology Specialist: Joe Vetere Rights and Permissions Advisor: Michael Joyce Manufacturing Buyer: Debbie Rossi Cover Image: Fotolia: Yurok Aleksandrovich Creative Director: Jayne Conte Designer: Suzanne Behnke Many of the designations used by manufacturers and sellers to distinguish their products are claimed as trademarks. Where those designations appear in this book, and Pearson Education was aware of a trademark claim, the designations have been printed in initial caps or all caps. Library of Congress Cataloging-in-Publications Data Hogg, Robert V. Introduction to mathematical statistics / Robert V. Hogg, Joseph W. McKean, Allen T. Craig. – 7th ed. p. cm. ISBN 978-0-321-79543-4 1. Mathematical statistics. I. McKean, Joseph W., 1944- II. Craig, Allen T. (Allen Thorton), 1905- III. Title. QA276.H59 2013 519.5–dc23 2011034906 Copyright 2013, 2005, 1995 Pearson Education, Inc. All rights reserved. No part of this publication may be reproduced, stored in a re- trieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Printed in the United States of America . For information on obtain- ing permission for use of material in this work, please submit a written request to Pearson Education, Inc., Rights and Contracts Department, 501 Boylston Street, Suite 900 , Boston , MA 02116 , fax your request to 617-671-3447, or e-mail at http://www.pearsoned.com/legal/permissions.htm. 1 2 3 4 5 6 7 8 9 10CRS15 14 13 12 11 www.pearsonhighered.com ISBN 13: 978-0-321-79543-4 ISBN 10: 0-321-79543-1
    To Ann and to Marge
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    Contents Preface 1 Probability and Distributions 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 The Probability Set Function . . . . . . . . . . . . . . . . . . . . . . 1.4 Conditional Probability and Independence . . . . . . . . . . . . . . . 1.5 Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Discrete Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Continuous Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Expectation of a Random Variable . . . . . . . . . . . . . . . . . . . 1.9 Some Special Expectations . . . . . . . . . . . . . . . . . . . . . . . 1.10 Important Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.1 Transformations 1.7.1 Transformations ix 1 1 3 10 21 32 40 42 44 46 52 57 68 2 Multivariate Distributions 2.1 Distributions of Two Random Variables . . . . . . . . . . . . . . . . 2.1.1 Expectation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Transformations: Bivariate Random Variables . . . . . . . . . . . . . 2.3 Conditional Distributions and Expectations . . . . . . . . . . . . . . 2.4 The Correlation Coefficient 2.5 2.6 Extension to Several Random Variables 73 73 79 84 94 . . . . . . . . . . . . . . . . . . . . . . . 102 Independent Random Variables . . . . . . . . . . . . . . . . . . . . . 110 . . . . . . . . . . . . . . . . 117 Multivariate Variance-Covariance Matrix . . . . . . . . . . . 123 2.7 Transformations for Several Random Variables . . . . . . . . . . . . 126 2.8 Linear Combinations of Random Variables . . . . . . . . . . . . . . . 134 2.6.1 ∗ 3 Some Special Distributions 139 3.1 The Binomial and Related Distributions . . . . . . . . . . . . . . . . 139 3.2 The Poisson Distribution . . . . . . . . . . . . . . . . . . . . . . . . 150 3.3 The Γ, χ2, and β Distributions . . . . . . . . . . . . . . . . . . . . . 156 3.4 The Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . 168 . . . . . . . . . . . . . . . . . . . . . 174 3.4.1 Contaminated Normals v
    vi Contents ∗ 3.6 3.5 The Multivariate Normal Distribution . . . . . . . . . . . . . . . . . 178 3.5.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 t- and F -Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 189 3.6.1 The t-distribution . . . . . . . . . . . . . . . . . . . . . . . . 189 3.6.2 The F -distribution . . . . . . . . . . . . . . . . . . . . . . . . 191 Student’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . 193 3.6.3 . . . . . . . . . . . . . . . . . . . . . . . . . . 197 3.7 Mixture Distributions 4 Some Elementary Statistical Inferences 4.1 Sampling and Statistics 203 . . . . . . . . . . . . . . . . . . . . . . . . . 203 4.1.1 Histogram Estimates of pmfs and pdfs . . . . . . . . . . . . . 207 4.2 Confidence Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 4.2.1 Confidence Intervals for Difference in Means . . . . . . . . . . 217 4.2.2 Confidence Interval for Difference in Proportions . . . . . . . 219 . . . . 223 4.3 Confidence Intervals for Parameters of Discrete Distributions 4.4 Order Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 4.4.1 Quantiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 4.4.2 Confidence Intervals for Quantiles . . . . . . . . . . . . . . . 234 4.5 Introduction to Hypothesis Testing . . . . . . . . . . . . . . . . . . . 240 4.6 Additional Comments About Statistical Tests . . . . . . . . . . . . . 248 4.7 Chi-Square Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 4.8 The Method of Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . 261 4.8.1 Accept–Reject Generation Algorithm . . . . . . . . . . . . . . 268 . . . . . . . . . . . . . . . . . . . . . . . . . . 273 4.9.1 Percentile Bootstrap Confidence Intervals . . . . . . . . . . . 273 4.9.2 Bootstrap Testing Procedures . . . . . . . . . . . . . . . . . . 276 ∗ Tolerance Limits for Distributions . . . . . . . . . . . . . . . . . . . 284 4.9 Bootstrap Procedures 4.10 5 Consistency and Limiting Distributions 289 5.1 Convergence in Probability . . . . . . . . . . . . . . . . . . . . . . . 289 5.2 Convergence in Distribution . . . . . . . . . . . . . . . . . . . . . . . 294 5.2.1 Bounded in Probability . . . . . . . . . . . . . . . . . . . . . 300 5.2.2 Δ-Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 5.2.3 Moment Generating Function Technique . . . . . . . . . . . . 303 5.3 Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 307 . . . . . . . . . . . . . . . 314 5.4 Extensions to Multivariate Distributions ∗ 6 Maximum Likelihood Methods 321 6.1 Maximum Likelihood Estimation . . . . . . . . . . . . . . . . . . . . 321 6.2 Rao–Cram´er Lower Bound and Efficiency . . . . . . . . . . . . . . . 327 6.3 Maximum Likelihood Tests . . . . . . . . . . . . . . . . . . . . . . . 341 6.4 Multiparameter Case: Estimation . . . . . . . . . . . . . . . . . . . . 350 6.5 Multiparameter Case: Testing . . . . . . . . . . . . . . . . . . . . . . 359 6.6 The EM Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367
    Contents vii 7 Sufficiency 375 7.1 Measures of Quality of Estimators . . . . . . . . . . . . . . . . . . . 375 7.2 A Sufficient Statistic for a Parameter . . . . . . . . . . . . . . . . . . 381 7.3 Properties of a Sufficient Statistic . . . . . . . . . . . . . . . . . . . . 388 7.4 Completeness and Uniqueness . . . . . . . . . . . . . . . . . . . . . . 392 7.5 The Exponential Class of Distributions . . . . . . . . . . . . . . . . . 397 7.6 Functions of a Parameter . . . . . . . . . . . . . . . . . . . . . . . . 402 7.7 The Case of Several Parameters . . . . . . . . . . . . . . . . . . . . . 407 7.8 Minimal Sufficiency and Ancillary Statistics . . . . . . . . . . . . . . 415 7.9 Sufficiency, Completeness, and Independence . . . . . . . . . . . . . 421 8 Optimal Tests of Hypotheses 429 8.1 Most Powerful Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 8.2 Uniformly Most Powerful Tests . . . . . . . . . . . . . . . . . . . . . 439 8.3 Likelihood Ratio Tests . . . . . . . . . . . . . . . . . . . . . . . . . . 447 8.4 The Sequential Probability Ratio Test . . . . . . . . . . . . . . . . . 459 8.5 Minimax and Classification Procedures . . . . . . . . . . . . . . . . . 466 8.5.1 Minimax Procedures . . . . . . . . . . . . . . . . . . . . . . . 466 8.5.2 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . 469 9 Inferences About Normal Models 473 9.1 Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473 9.2 One-Way ANOVA . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478 9.3 Noncentral χ2 and F -Distributions . . . . . . . . . . . . . . . . . . . 484 9.4 Multiple Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . 486 9.5 The Analysis of Variance . . . . . . . . . . . . . . . . . . . . . . . . 490 9.6 A Regression Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 497 9.7 A Test of Independence . . . . . . . . . . . . . . . . . . . . . . . . . 506 9.8 The Distributions of Certain Quadratic Forms . . . . . . . . . . . . . 509 9.9 The Independence of Certain Quadratic Forms . . . . . . . . . . . . 516 10 Nonparametric and Robust Statistics 525 10.1 Location Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525 10.2 Sample Median and the Sign Test . . . . . . . . . . . . . . . . . . . . 528 10.2.1 Asymptotic Relative Efficiency . . . . . . . . . . . . . . . . . 533 10.2.2 Estimating Equations Based on the Sign Test . . . . . . . . . 538 10.2.3 Confidence Interval for the Median . . . . . . . . . . . . . . . 539 10.3 Signed-Rank Wilcoxon . . . . . . . . . . . . . . . . . . . . . . . . . . 541 10.3.1 Asymptotic Relative Efficiency . . . . . . . . . . . . . . . . . 546 10.3.2 Estimating Equations Based on Signed-Rank Wilcoxon . . . 549 10.3.3 Confidence Interval for the Median . . . . . . . . . . . . . . . 549 10.4 Mann–Whitney–Wilcoxon Procedure . . . . . . . . . . . . . . . . . . 551 10.4.1 Asymptotic Relative Efficiency . . . . . . . . . . . . . . . . . 555 10.4.2 Estimating Equations Based on the Mann–Whitney–Wilcoxon 556 10.4.3 Confidence Interval for the Shift Parameter Δ . . . . . . . . . 557 10.5 General Rank Scores . . . . . . . . . . . . . . . . . . . . . . . . . . . 559
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