Deshpande J. V. - Nonparametric Statistics: Theory And Methods

NONPARAMETRIC STATISTICS

THEORY AND METHODS

NONPARAMETRIC STATISTICS

THEORY AND METHODS

Jayant V Deshpande

University of Pune, India

Uttara Naik-Nimbalkar

University of Pune, India

Isha Dewan

Indian Statistical Institute, India

Published by

World Scientific Publishing Co. Pte. Ltd.

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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 Design of Experiments, or Regression Analysis. We feel that there is a need for an alternative text book for students which can also be a reference book 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 distribution theory, relative efficiency, etc. We also include some examples to illustrate the methodology and some exercises for the students. We have not included any tables of critical points as they are now generally available in common software packages along with programs to calculate the statistics. 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 comments which were useful in improving the chapter.

We thank our respective families for their support during this work and the authorities of the various institutes where we worked in the last few years.

Jayant V. Deshpande

University of Pune

Uttara V. Naik-Nimbalkar

University of Pune & now at Indian Institute of Science Education and Research, Pune

Isha Dewan

Indian Statistical Institute, New Delhi

Statistical inference is the methodology to discover statistical laws regarding the outcomes of random experiments and phenomena. Randomness is prevalent in many aspects of everyday experience. Whether it will rain or not on a particular day, the price of a particular stock on the stock market, the outcome of the toss of a coin, the weight of a newly born baby, the increase in milk yield of a cow by adding certain supplements to its feed, the time required for a chemical reaction to be completed in the presence of a fixed level of a catalytic agent, the number of emissions from a radio-active source in a fixed interval of time, the time for which an electric bulb will function before it fuses, are all unpredictable prior to the event. Some of the above may be regarded as outcomes of controlled experiments whereas others are observations on uncontrolled phenomena. However the common theme of unpredictability, i.e., randomness of outcomes runs through all of them. It is impossible to exactly predict the outcome of the next trial of any of the above or similar cases.

All the same, it is our common experience that these outcomes of random trials exhibit certain patterns of regularity in long sequences. These patterns of regularity are called statistical laws governing the outcomes. For example, if the same coin is tossed again and again, the ratio of number of heads to the total number of tosses seems to stabilize to a number p (0 p 1) as the number of tosses increases indefinitely. We call the coin fair or unbiased if p happens to be 1/2. This number p, which may be thought to be intrinsic to the coin in question, is called the probability of obtaining a head at a trial and may be said to be the statistical law governing the outcomes of the random experiment consisting of tossing this coin. In general, the purpose of statistical inference is to discover the statistical laws governing the outcomes of a random experiment by repeated performance of the experiment or by making repeated observations of a phenomenon.

In order to make progress in the analysis of the situation we have to introduce and impose certain useful mathematical structure on the random experiment being conducted or the random phenomena being observed.

For any random experiment under study we assume that the set of all possible outcomes is well defined and known. For example, the set for tossing a coin consists of two outcomes only: {H, T}, head and tail; the set for counting the number of emissions from radio active source is {0, 1, 2, } the class of all non-negative integers without a finite upper bound, the set for the life of an electric bulb is half of the real line [0, ), again without a finite upper bound. This set of all possible outcomes is called the sample space of the random experiment. Sometimes, for the sake of convenience the sample space assumed for a random experiment might be somewhat different from strict reality; in the coin tossing experiment standing on edge, losing the coin, etc. are not taken into account. Also with measuring quantities like weight, length, time, etc. the actual observations are accurate only to the extent of the smallest possible unit that can be recorded by the measuring instrument, say, one gram, one millimetre, one second, etc. But again for the sake of convenience continuous intervals are accepted as the sample spaces. Such simplifications are often carried out wherever no ambiguity is involved.