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1 This page intentionally left blank IntroductiontoMathematical StatisticsEighth EditionRobert V. HoggUniversity of IowaJoseph W. McKeanWestern Michigan UniversityAllen T. CraigLate Professor of StatisticsUniversity of IowaDirector, Portfolio Management: Deirdre LynchCourseware Portfolio Manager: Patrick BarberaPortfolio Management Assistant: Morgan DannaContent Producer: Lauren MorseManaging Producer: Scott DisannoProduct Marketing Manager: Yvonne VannattaField Marketing Manager: Evan St. CyrMarketing Assistant: Jon BryantSenior Author Support/Technology Specialist: Joe VetereManager, Rights and Permissions: Gina CheselkaManufacturing Buyer: Carol Melville, LSC CommunicationsArt Director: Barbara AtkinsonProduction Coordination and Illustrations: IntegraCover Design: Studio MontageCover Image: Aleksandarvelasevic/Digital Vision Vectors/Getty 2019, 2013, 2005 by Pearson Education, Inc.

2 All Rights Reserved. Printed in theUnited States of America. This publication is protected by copyright, and permission shouldbe obtained from the publisher prior to any prohibited reproduction, storage in a retrieval sys-tem, or transmission in any form or by any means, electronic, mechanical, photocopying, record-ing, or otherwise. For information regarding permissions, request forms and the appropriatecontacts within the Pearson Education Global Rights & Permissions department, please and ALWAYS LEARNING are exclusive trademarks owned by Pearson Education,Inc. or its affiliates in the and/or other countries. Unless otherwise indicated herein, anythird-party trademarks that may appear in this work are the property of their respective ownersand any references to third-party trademarks, logos or other trade dress are for demonstrative ordescriptive purposes only.

3 Such references are not intended to imply any sponsorship, endorsement,authorization, or promotion of Pearson s products by the owners of such marks, or any relationshipbetween the owner and Pearson Education, Inc. or its affiliates, authors, licensees or of Congress Cataloging-in-Publications DataNames: Hogg, Robert V., author.|McKean, Joseph W., 1944- author.|Craig,Allen T. (Allen Thornton), 1905- : Introduction to mathematical statistics / Robert V. Hogg, Late Professor of Statistics,University of Iowa, Joseph W. McKean, Western Michigan University, Allen T. Craig,Late Professor of Statistics, University of : Eighth edition.|Boston : Pearson, [2019]|Includesbibliographical references and : LCCN 2017033015|ISBN 9780134686998|ISBN 0134686993 Subjects: LCSH: Mathematical : LCC QA276.

4 H59 2019|DDC dc23 LC record available 13: 978-0-13-468699-8 ISBN 10: 0-13-468699-3 Dedicated to my wife Margeand to the memory of Bob HoggThis page intentionally left blank ContentsPrefacexi1 Probability and Sets .. ReviewofSetTheory .. The Probability Set Function .. Additional Properties of Probability .. Conditional Probability and Independence .. RandomVariables .. DiscreteRandomVariables .. Transformations .. ContinuousRandomVariables .. Quantiles .. Transformations .. Mixtures of Discrete and Continuous Type Distributions .. ExpectationofaRandomVariable.

5 R Computation for an Estimation of the Expected Gain .. SomeSpecialExpectations .. 782 Multivariate DistributionsofTwoRandomVariables .. Conditional Distributions and Expectations .. TheCorrelationCoefficient .. ExtensiontoSeveralRandomVariables .. Transformations for Several Random Variables .. 1513 Some Special Negative Binomial and Geometric Distributions .. MultinomialDistribution .. HypergeometricDistribution .. ThePoissonDistribution .. The , 2,and Distributions .. The 2-Distribution .. The TheMultivariateNormalDistribution.

6 Multivariate Normal Distribution, General Case .. Thet-distribution .. Student 2184 Some Elementary Statistical SamplingandStatistics .. Confidence Intervals .. Confidence Intervals for Difference in Means .. Confidence Interval for Difference in Proportions .. Confidence Intervals for Parameters of Discrete Distributions .. Quantiles .. Confidence Intervals for Quantiles .. Additional Comments About Statistical Tests .. Observed Significance Level,p-value .. Chi-SquareTests .. Accept Reject Generation Algorithm .. BootstrapProcedures.

7 Percentile Bootstrap Confidence Intervals .. 315 Contentsvii5 Consistency and Limiting Convergence in Probability .. SamplingandStatistics .. Bounded in Probability .. CentralLimitTheorem .. ExtensionstoMultivariateDistributions .. 3486 Maximum Likelihood MaximumLikelihoodEstimation .. Rao Cram erLowerBoundandEfficiency .. MaximumLikelihoodTests .. 4047 MeasuresofQualityofEstimators .. FunctionsofaParameter .. BootstrapStandardErrors .. Minimal Sufficiency and Ancillary Statistics .. Sufficiency, Completeness, and Independence.

8 4618 Optimal Tests of MostPowerfulTests .. UniformlyMostPowerfulTests .. Likelihood Ratio Tests for Testing Means of Normal Distri-butions .. Likelihood Ratio Tests for Testing Variances of Normal Dis-tributions .. The Sequential Probability Ratio Test .. MinimaxandClassificationProcedures .. Classification .. 510viiiContents9 Inferences About Normal Linear One-WayANOVA .. Noncentral MultipleComparisons .. Two-WayANOVA .. ARegressionProblem .. GeometryoftheLeastSquaresFit .. ATestofIndependence .. The Distributions of Certain Quadratic Forms .. The Independence of Certain Quadratic Forms.

9 56210 Nonparametric and Robust .. AsymptoticRelativeEfficiency .. Estimating Equations Based on the Sign Test .. Confidence Interval for the Median .. AsymptoticRelativeEfficiency .. Estimating Equations Based on Signed-Rank Wilcoxon .. Confidence Interval for the Median .. Whitney AsymptoticRelativeEfficiency .. Estimating Equations Based on the Mann Whitney Wilcoxon Confidence Interval for the Shift Parameter .. Monte Carlo Investigation of Power .. GeneralRankScores .. Efficacy .. Estimating Equations Based on General Scores .. AdaptiveProcedures .. Kendall s.

10 Spearman sRho .. 645 Contentsix11 Bayesian .. Prior and Posterior Distributions .. BayesianPointEstimation .. BayesianIntervalEstimation .. BayesianTestingProcedures .. EmpiricalBayes .. 682A Mathematical RegularityConditions .. Sequences .. 688B R Basics .. Probability Distributions .. Loops .. Input and Output .. 700C Lists of Common Distributions703D Tables of Distributions707E References715F answers to selected Exercises721 Index733 This page intentionally left blank PrefaceWe have made substantial changes in this edition ofIntroduction to MathematicalStatistics. Some of these changes help students appreciate the connection betweenstatistical theory and statistical practice while other changes enhance the develop-ment and discussion of the statistical theory presented in this of the changes in this edition reflect comments made by our readers.