Examples in Parametric Inference with R

475 Pages·2016·3.969 MB·English

by  Ulhas Jayram Dixit (auth.)

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
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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
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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
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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.
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