Nonparametric Statistics: Theory and Methods

275 Pages·2017·3.534 MB·English

by  Jayant V Deshpande, Uttara Naik-Nimbalkar, Isha Dewan

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Published by
World Scientific Publishing Co. Pte. Ltd.
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USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress Cataloging-in-Publication Data
Names: Deshpande, J. V., author. | Naik-Nimbalkar, Uttara, author. | Dewan, Isha, author.
Title: Nonparametric statistics : theory and methods / by Jayant V. Deshpande
(University of Pune, India), Uttara Naik-Nimbalkar (University of Pune, India),
Isha Dewan (Indian Statistical Institute, India).
Description: New Jersey : World Scientific, 2017. | Includes bibliographical references and index.
Identifiers: LCCN 2017029415 | ISBN 9789814663571 (hardcover : alk. paper)
Subjects: LCSH: Nonparametric statistics. | Distribution (Probability theory)
Classification: LCC QA278.8 .D37 2017 | DDC 519.5/4--dc23
LC record available at https://lccn.loc.gov/2017029415
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Copyright © 2018 by World Scientific Publishing Co. Pte. Ltd.
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PREFACE
Books on Nonparametric Statistics are not as numerous as, say, those on
DesignofExperiments,orRegressionAnalysis. Wefeelthatthereisaneed
foranalternativetextbookforstudentswhichcanalsobeareferencebook
for practitioners of statistical methods. The present book is offered with
this need in view. The emphasis here is on the heuristic and theoretical
base of the subject along with the usefulness of Nonparametric Methods
in various situations. The audience we have in mind is that of advanced
undergraduate and graduate students along with users of these methods.
The first chapter is an Introduction to Statistical Inference in general
with the role of Nonparametric Statistics within it. The second to ninth
chapters deal with classical methods, and the last three chapters deal with
more computation intensive methods which are often only asymptotically
nonparametric. The book ends with an Appendix which brings together
many of the probabilistic results required to prove the asymptotic distri-
bution theory, relative efficiency, etc. We also include some examples to
illustrate the methodology and some exercises for the students. We have
notincludedanytablesofcriticalpointsastheyarenowgenerallyavailable
in common software packages along with programs to calculate the statis-
tics. At many places we advocate the use of the public domain software
package R.
We have extensive experience in teaching such courses, in developing
such methodology and also applying it in practice. We hope that the road
map we provide here is effective towards these three aims.
Prof R.V.Ramamoorthy read an earlier version of the chapter on
Bayesian Nonparametric Methods and we gratefully acknowledge his com-
ments which were useful in improving the chapter.
v
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vi NONPARAMETRIC STATISTICS: THEORY AND METHODS
Wethankourrespectivefamiliesfortheirsupportduringthisworkand
the authorities of the various institutes where we worked in the last few
years.
Jayant V. Deshpande
University of Pune
Uttara V. Naik-Nimbalkar
UniversityofPune&nowatIndianInstituteofScienceEducationandResearch,
Pune
Isha Dewan
Indian Statistical Institute, New Delhi
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CONTENTS
PREFACE v
1. PRINCIPLES OF STATISTICAL INFERENCE 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Mathematical Structure on Random Experiments . . . . . 2
1.3 Random Variables and their Probability Distributions . . 4
1.4 Parametric and Nonparametric Statistical Models . . . . . 6
1.5 Estimation of Parameters . . . . . . . . . . . . . . . . . . 8
1.6 Maximum Likelihood Estimation . . . . . . . . . . . . . . 14
1.7 Observation with Censored Data . . . . . . . . . . . . . . 16
1.7.1 Type I (Time) Censoring . . . . . . . . . . . . . . 16
1.7.2 Type II (Order) Censoring . . . . . . . . . . . . . 16
1.7.3 Type III (Random Censoring) . . . . . . . . . . . 17
1.8 Ranked Set Sampling . . . . . . . . . . . . . . . . . . . . . 18
1.8.1 Estimation of the Mean . . . . . . . . . . . . . . . 19
1.9 Introduction to Bayesian Inference . . . . . . . . . . . . . 20
1.10 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2. ORDER STATISTICS 23
2.1 Random Sample from a Continuous Distribution . . . . . 23
2.2 Order Statistics . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3 Sampling Distribution of Order Statistics . . . . . . . . . 24
2.4 Sufficiency and Completeness of Order Statistics . . . . . 26
2.5 Probability Integral Transformation. . . . . . . . . . . . . 27
2.6 Order Statistics of Uniform and Exponential Random
Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
vii
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viii NONPARAMETRIC STATISTICS: THEORY AND METHODS
2.6.1 Uniform Distribution . . . . . . . . . . . . . . . . 29
2.6.2 Exponential Distribution . . . . . . . . . . . . . . 30
2.7 Confidence Intervals for Population Quantiles Based on
Simple Random Sampling . . . . . . . . . . . . . . . . . . 32
2.8 Confidence Intervals of the Population Quantiles Based on
Ranked Set Sampling . . . . . . . . . . . . . . . . . . . . . 34
2.9 Tolerance Intervals . . . . . . . . . . . . . . . . . . . . . . 36
2.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3. EMPIRICAL DISTRIBUTION FUNCTION 39
3.1 Inference in the Nonparametric Setting . . . . . . . . . . . 39
3.2 The Empirical Distribution Function . . . . . . . . . . . . 40
3.3 Properties of the Empirical Distribution Function . . . . . 40
3.4 The M.L.E. of the Distribution Function . . . . . . . . . . 43
3.5 Confidence Intervals for the Distribution Function . . . . 43
3.6 Actuarial Estimator of the Survival Function . . . . . . . 44
3.7 Kaplan-Meier Estimator of the Distribution Function . . . 46
3.8 The Nelson-Aalen Estimator of the Cumulative Hazard
Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.9 The Nonparametric Bootstrap . . . . . . . . . . . . . . . . 53
3.9.1 Bootstrap Confidence Interval Based on Normal
Approximation . . . . . . . . . . . . . . . . . . . . 54
3.9.2 Bootstrap Percentile Intervals . . . . . . . . . . . 54
3.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4. THE GOODNESS OF FIT PROBLEM 57
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.2 Chi-squared Goodness of Fit Test . . . . . . . . . . . . . . 57
4.2.1 Completely Specified Distribution Function . . . 58
4.2.2 Some Parameters of the Distribution Function are
Unknown . . . . . . . . . . . . . . . . . . . . . . . 60
4.3 The Kolmogorov-Smirnov Goodness of Fit Test . . . . . . 63
4.3.1 Null Distribution of the Statistic . . . . . . . . . . 64
4.3.2 Confidence Bands for F(x) Based on Complete
Samples . . . . . . . . . . . . . . . . . . . . . . . . 65
4.3.3 Confidence Bands for Survival Function Based on
Censored Random Samples . . . . . . . . . . . . . 67
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CONTENTS ix
4.3.4 Comparison of the Chi-square and Kolmogorov
Tests for the Goodness of Fit Hypotheses . . . . . 69
4.4 Tests of Exponentiality . . . . . . . . . . . . . . . . . . . . 72
4.4.1 The Hollander-Proschan (1972) Test . . . . . . . . 73
4.4.2 The Deshpande (1983) Test . . . . . . . . . . . . . 74
4.5 Tests for Normality . . . . . . . . . . . . . . . . . . . . . . 75
4.5.1 The Shapiro-Wilk-Francia-D’Agostino Tests . . . 76
4.6 Diagnostic Methods for Identifying the Family of
Distribution Functions . . . . . . . . . . . . . . . . . . . . 77
4.6.1 The Q-Q Plot . . . . . . . . . . . . . . . . . . . . 77
4.6.2 The log Q-Q Plot . . . . . . . . . . . . . . . . . . 78
4.6.3 The P-P Plot . . . . . . . . . . . . . . . . . . . . . 79
4.6.4 The T-T-T Plot . . . . . . . . . . . . . . . . . . . 80
5. THE ONE SAMPLE PROBLEM 87
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.2 The Sign Test for a Specified Value of the Median. . . . . 87
5.3 Wilcoxon Signed Rank Test . . . . . . . . . . . . . . . . . 90
5.4 Ranked Set Sampling Version of the Sign Test . . . . . . . 93
5.5 RankedSetSamplingVersionoftheWilcoxonSignedRank
Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.6 Tests for Randomness . . . . . . . . . . . . . . . . . . . . 96
5.7 Nonparametric Confidence Interval for the Median of a
Symmetric Distribution Based on the Signed Rank
Statistic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6. OPTIMAL NONPARAMETRIC TESTS 103
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.2 Most Powerful Rank Tests Against a Simple Alternative . 104
6.3 Locally Most Powerful (LMP) Rank Tests . . . . . . . . . 106
6.4 Rank Statistics and Its Null Distribution . . . . . . . . . . 107
6.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
7. THE TWO SAMPLE PROBLEM 111
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 111
7.2 The Location Problem . . . . . . . . . . . . . . . . . . . . 112
7.2.1 The Wilcoxon Rank Sum Test . . . . . . . . . . . 113