Distribution-Free Statistical Methods

272 Pages·1981·7.158 MB·English

by  J. S. Maritz (auth.)

MONOGRAPHS ON
APPLIED PROBABILITY AND STATISTICS
DISTRIBUTION-FREE STATISTICAL METHODS
MONOGRAPHS ON
APPLIED PROBABILITY AND STATISTICS
General Editor
D.R. COX, F.R.S.
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Distribution-Free
Statistical Methods
J.S. MARITZ
Department of Mathematical Statistics
La Trobe University
Australia
SPRINGER-SCIENCE+BUSINESS MEDIA B.V.
© 1981 1.S. Maritz
Originally published by Chapman & Hall in 1981
Softcover reprint of the hardcover 1s t edition 1981
All rights reserved. No part of
this book may be reprinted, or reproduced
or utilized in any form or by any electronic,
mechanical or other means, now known or hereafter
invented, including photocopying and recording,
or in any information storage and retrieval
system, without permission in writing
from the Publisher.
British Library Cataloguing in Publication Data
Maritz, 1. S.
Distribution-free statistical methods. - (Monographs on
applied probability and statistics)
1. Probabilities 2. Mathematical statistics
I. Title II. Series
519.2 QA278 80-42136
ISBN 978-0-412-15940-4 ISBN 978-1-4899-3302-7 (eBook)
DOI 10.1007/978-1-4899-3302-7
Contents
Preface Page ix
1 Basic concepts in distribution-free methods 1
1.1 Introduction 1
1.2 Randomization and exact tests 2
1.3 Consistency of tests 5
1.4 Point estimation of a single parameter 7
1.5 Confidence limits 7
1.6 Efficiency considerations in the one-parameter
case 8
1.7 Multiple samples and parameters 11
1.8 Normal approximations 15
2 One-sample location problems 21
2.1 Introduction 21
2.2 The median 23
2.3 Symmetric distributions 28
2.4 Asymmetric distributions: M-estimates 58
3 Miscellaneous one-sample problems 61
3.1 Introduction 61
3.2 Dispersion: the interquartile range 61
3.3 The sample distribution function and related
inference 65
3.4 Estimation of F when some observations are
censored 73
4 Two-sample problems 81
4.1 Types of two-sample problems 81
4.2 The basic randomization argument 82
4.3 Inference about location difference 83
4.4 Proportional hazards (Lehmann alternative) 108
4.5 Dispersion alternatives 117
viii CONTENTS
5 Straight-line regression 123
5.1 The model and some preliminaries 123
5.2 Inference about f3 only 124
5.3 Joint inference about IX and P 152
6 Multiple regression and general linear models 173
6.1 Introduction 173
6.2 Plane regression: two independent variables 173
6.3 General linear models 202
6.4 One-way analysis of variance 206
6.5 Two-way analysis of variance 209
7 Bivariate problems 215
7.1 Introduction 215
7.2 Tests of correlation 215
7.3 One-sample location 220
7.4 Two-sample location problems 236
7.5 Three-sample location problems 245
7.6 Multiple-sample location problems 251
Appendix A 253
Appendix B 257
Bibliography 259
Subject index 263
Author index 265
Preface
The preparation of several short courses on distribution-free statisti
cal methods for students at third and fourth year level in Australian
universities led to the writing ofthis book. My criteria for the courses
were, firstly, that the subject should have a clearly recognizable
underlying common thread rather than appear to be a collection of
isolated techniques. Secondly, some discussion of efficiency seemed
essential, at a level where the students could appreciate the reasons for
the types of calculations that are performed, and be able actually to do
some of them. Thirdly, it seemed desirable to emphasize point and
interval estimation rather more strongly than is the case in many of
the fairly elementary books in this field.
Randomization, or permutation, is the fundamental idea that
connects almost all of the methods discussed in this book.
Application of randomization techniques to original observations, or
simple transformations of the observations, leads generally to
conditionally distribution-free inference. Certain transformations,
notably 'sign' and 'rank' transformations may lead to unconditionally
distribution-free inference. An attendant advantage is that useful
tabulations of null distributions of test statistics can be produced.
In my experience students find the notion of asymptotic relative
efficiency of testing difficult. Therefore it seemed worthwhile to give a
rather informal introduction to the relevant ideas and to concentrate
on the Pitman 'efficacy' as a measure of efficiency.
Most of the impetus to use distribution-free methods was originally
in hypothesis testing. It is now well recognized that adaptation of
some of the ideas to point estimation can be advantageous from the
points of view of efficiency and robustness. Pedagogically there are
also advantages in emphasizing estimation. One of them is that one
can adopt the straightforward approach of defining relative efficiency
in terms of variances of estimates. Another is that using the notion of
an estimating equation makes it easy to relate the distribution-free
techniques to methods which will have been encountered in the
x PREFACE
standard statistics courses. Examples include the method of mo
ments, and large sample approximations to standard errors of
estimates.
The aim of this book is to give an introduction to the distribution
free way of thinking and sufficient detail of some standard techniques
to be useful as a practical guide. It is not intended as a compendium of
distribution-free techniques and some readers may find that a
technique which they regard as important is not mentioned. For the
most part the book deals with problems oflocation and location shift.
They include one- and two-sample location problems, and some
aspects of regression and of the 'analysis of variance'.
Although some ofthe presentation is somewhat different from what
appears to have become the standard in this field, very little, if any, of
the material is original. Much has been gleaned from various texts.
Direct acknowledgement of my indebtedness to the authors of these
works is made by the listing of general references. Through these, and
other references, I also acknowledge indirectly the work of other
authors whose names may not appear in the bibliography. No serious
attempt has been made to attribute ideas to their originators. Specific
references are given only where it is felt that readers may be
particularly interested in more detail.
While the origins of this book are in undergraduate teaching I do
hope that some experienced statisticians will find parts of it interest
ing. In particular, developments in point and interval estimation, and
noting of their connections with 'robust' methods have taken place
fairly recently. Some interesting problems of estimating standard
errors, as yet not fully resolved, are touched upon in several places.
Many of my colleagues have helped me, by discussion and by
reading sections of manuscript. Dr D.G. Kildea read the first draft of
Chapter 2 and his detailed comments led to many improvements.
Dr B.M. Brown was not only a patient listener on many occasions
but also generously provided Appendix A.
Melbourne, November 1980 lS. Maritz