ebook img

Examples in Parametric Inference with R PDF

475 Pages·2016·3.969 MB·English
Save to my drive
Quick download
Download
Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.

Preview Examples in Parametric Inference with R

Ulhas Jayram Dixit Examples in Parametric Inference with R Examples in Parametric Inference with R Ulhas Jayram Dixit Examples in Parametric Inference with R 123 UlhasJayram Dixit Department ofStatistics University of Bombay Mumbai,Maharashtra India ISBN978-981-10-0888-7 ISBN978-981-10-0889-4 (eBook) DOI 10.1007/978-981-10-0889-4 LibraryofCongressControlNumber:2016936960 ©SpringerScience+BusinessMediaSingapore2016 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor foranyerrorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerScience+BusinessMediaSingaporePteLtd. To my parents late Shri Jayram Shankar Dixit alias Appa and late Shrimati Kamal J. Dixit alias Yammi Preface Thisbookoriginallygrewoutofmynotesonthestatisticalinferencecoursesatthe DepartmentofStatistics,UniversityofMumbai.Ihaveexperiencedthatreasonably good M.Sc. (Statistics) students are many a time not able to understand or solve problems from some available texts on statistical inference. These books are excellent in terms of content, but the presentation is highly sophisticated. For instance, proofs of various theorems are given in brief and a few examples are provided.Toovercomethisdifficulty,Ihavesolvedmanyexamplesand,wherever necessary, a program in R is also given. Further, important proofs in this book are presented in such a manner that they are easy to understand. Through this book, we expect students to know matrix algebra, calculus, probabilitytheory,anddistributiontheory.Thisbookwillserveasanexcellenttool for teaching statistical inference courses. The book consists of many solved and unsolved problems. Instructors can assign homework problems from the exercises andstudentswillfindthesolvedexampleshugelybeneficialinsolvingtheexercise problems. In “Prerequisite”, we have discussed some basic concepts like distribution function and order statistics and illustrated them by using interesting examples. Chapter1dealswithsufficiencyandcompleteness.Inthischapter,wehavesolved 37 examples. Chapter 2 deals with unbiased estimation. In the last 30 years of my teaching,Ifoundthatstudentswerealwaysconfusedabouttherelationshipbetween sufficiency and unbiasedness. We have explained this relationship with various examplesinthischapter.Chapter3isdevotedtomethodofmomentsandmaximum likelihood. In Chap. 4, we deal with lower bound for the variance of an unbiased estimator. Popular concepts like Cramer–Rao (1945, 1946) and Bhattacharya (1946, 1950) lower bound are discussed in detail. Chapter 4 also deals with ChapmanandRobbins(1951)andKiefer(1952)lowerboundforthevarianceofan estimate but does not require regularity conditions. In Chap. 5, the concept of consistency is discussed in detail and illustrated by using different examples. In Chap.6,Bayesianestimationisbrieflydiscussed.Chapters7and8aresignificantly large chapters. Testing of hypothesis is studied in Chap. 7, whereas unbiased and vii viii Preface other tests are studied in Chap. 8. We have given R programs in various chapters. No originality is claimed except perhaps in the presentation of the material. It will prove difficult to thank all my friends who have contributed in some or other way to make this book a reality. I am thankful to Prof. R.B. Bapat for his valuable suggestions to improve upon the content and presentation of the book. I also thank Dr. T.V. Ramanathan for making some valuable suggestions. I am thankful to Shamim Ahmad, senior editor at Springer India for encouraging me to publish this book through Springer and making it easy to go through the process. IthankProf. SeemaC.forreadingthebookforlanguage.Iamequallythankfulto Dr. Alok Dabade, Prof. Shailaja Kelkar, Dr. Mehdi Jabbari Nooghabi, Prof. S.AnnapurnaandProf.MandarBhanusheforvariousacademicdiscussionsrelated to the book and drawing figures. I am also very thankful to my son Anand and daughterVaidehiwhohelpedmesolvevariousproblems.Further,Iamthankfulto my wife Dr. (Mrs.) Vaijayanti for the insightful discussions on our book. We are grateful to Prof. Y.S. Sathe and Late Prof. M.N. Vartak for the diverse discussions which were helpful in understanding statistical inference. These dis- cussions were particularly helpful in solving problems on UMVUE and testing of hypotheses. We are thankful to Prof. B.V. Dhandra, Dr. D.B. Jadhav, and Prof. D.T. Jadhav for providing their M.Phil. dissertations. In spite of my best efforts, there might be some errors and misprints in the presentation. I owe these mistakes and request the readers to kindly bring them to my notice. Ulhas Jayram Dixit Contents 1 Sufficiency and Completeness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Sufficient Statistics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Minimal Sufficiency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.4 Ancillary Statistics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.5 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.6 Exponential Class Representation: Exponential Family. . . . . . . . . 31 1.7 Exercise 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2 Unbiased Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.1 Unbiased Estimates and Mean Square Error . . . . . . . . . . . . . . . . 39 2.2 Unbiasedness and Sufficiency . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.3 UMVUE in Nonexponential Families. . . . . . . . . . . . . . . . . . . . . 75 2.4 Exercise 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 3 Moment and Maximum Likelihood Estimators . . . . . . . . . . . . . . . . 109 3.1 Method of Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 3.2 Method of Maximum Likelihood. . . . . . . . . . . . . . . . . . . . . . . . 112 3.3 MLE in Censored Samples. . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 3.4 Newton–Raphson Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 3.5 Exercise 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 4 Bound for the Variance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 4.1 Cramme–Rao Lower Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 4.2 Bhattacharya Bound. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 4.3 Chapman-Robbins-Kiefer Bound. . . . . . . . . . . . . . . . . . . . . . . . 181 4.4 Exercise 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 ix x Contents 5 Consistent Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 5.1 Prerequisite Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 5.2 Definition and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 5.3 Consistent Estimator for Multiparameter. . . . . . . . . . . . . . . . . . . 207 5.4 Selection Between Consistent Estimators . . . . . . . . . . . . . . . . . . 210 5.5 Determination of n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 0 5.6 Exercise 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 6 Bayes Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 6.1 Bayes Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 6.2 Bayes Theorem for Random Variables. . . . . . . . . . . . . . . . . . . . 230 6.3 Bayesian Decision Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 6.4 Limit Superior and Limit Inferior . . . . . . . . . . . . . . . . . . . . . . . 237 6.5 Exercise 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 7 Most Powerful Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 7.1 Type-1 and Type-2 Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 7.2 Best Critical Region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 7.3 P-Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 7.4 Illustrative Examples on the MP Test. . . . . . . . . . . . . . . . . . . . . 285 7.5 Families with Monotone Likelihood Ratio . . . . . . . . . . . . . . . . . 320 7.6 Exercise 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344 8 Unbiased and Other Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 8.1 Generalized NP Lemma and UMPU Test. . . . . . . . . . . . . . . . . . 345 8.2 Locally Most Powerful Test (LMPT). . . . . . . . . . . . . . . . . . . . . 367 8.3 Similar Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 8.4 Neyman Structure Tests. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378 8.5 Likelihood Ratio Tests. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 8.6 Exercise 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 About the Author Ulhas Jayram Dixit is Professor, at the Department of Statistics, University of Mumbai, India. He is the first Rothamsted International Fellow at Rothamsted Experimental Station in the UK, which is the world’s oldest statistics department. Further, he received the Sesqui Centennial Excellence Award in research and teaching from the University of Mumbai in 2008. He is member of the New Zealand Statistical Association, the Indian Society for Probability and Statistics, Bombay Mathematical Colloquium, and the Indian Association for Productivity, Quality and Reliability. Editor of Statistical Inference and Design of Experiment (published by Narosa), Prof. Dixit has published over 40 papers in several inter- national journals of repute. His topics of interest are outliers, measure theory, distribution theory, estimation, elements of stochastic process, non-parametric inference, stochastic process, linear models, queuing and information theory, multivariate analysis, financial mathematics, statistical methods, design of experi- ments,andtestingofhypothesis.HereceivedhisPh.D.degreefromtheUniversity of Mumbai in 1989. xi

See more

The list of books you might like

Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.