Statistical Information and Likelihood: A Collection of Critical Essays by Dr. D. Basu PDF
Preview Statistical Information and Likelihood: A Collection of Critical Essays by Dr. D. Basu
Lecture Notes in Statistics Vol. 1: R.A. Fisher: An Appreciation. Edited by S.E. Fienberg and D.V. Hinkley. XI, 208 pages, 1980. Vol. 2: Mathematical Statistics and Probability Theory. Proceedings 1978. Edited by W. Klonecki, A. Kozek, and J. Rosinski. XXIV, 373 pages, 1980. Vol. 3: B. D. Spencer. Benefit-Cost Analysis of Data Used to Allocate Funds. VIII, 296 pages, 1980. Vol. 4: E. A. van Doorn, Stochastic Monotonicity and Queueing Applications of Birth-Death Proces ses. VI, 118 pages, 1981. Vol. 5: T. Rolski, Stationary Random Processes Associated with Point Processes. VI, 139 pages, 1981. Vol. 6: S. S. Gupta and D.-Y. Huang, Multiple Statistical Decision Theory: Recent Developments. VIII, 104 pages, 1981. Vol. 7: M. Akahira and K. Takeuchi, Asymptotic Efficiency of Statistical Estimators. VIII, 242 pages, 1981. Vol. 8: The First Pannonian Symposium on Mathematical Statistics. Edited by P. Revesz, L. Schmet terer, and V. M. Zolotarev. VI, 308 pages, 1981. Vol. 9: B. J0rgensen, Statistical Properties of the Generalized Inverse Gaussian Distribution. VI, 188 pages, 1981. Vol. 10: A. A. Mcintosh, Fitting Linear Models: An Application on Conjugate Gradient Algorithms. VI, 200 pages, 1982. Vol. 11: D. F. Nicholls and B. G. Quinn, Random Coefficient Autoregressive Models: An Introduction. V, 154 pages, 1982. Vol. 12: M. Jacobsen, Statistical Analysis of Counting Processes. VII, 226 pages, 1982. Vol. 13: J. Pfanzagl (with the assistance of W. Wefelmeyer), Contributions to a General Asymptotic Statistical Theory. VII, 315 pages, 1982. Vol. 14: GUM 82: Proceedings of the International Conference on Generalised Linear Models. Edited by R. Gilchrist. V, 188 pages, 1982. Vol. 15: K. R. W. Brewer and M. Hanif, Sampling with Unequal Probabilities. IX, 164 pages, 1983. Vol. 16: Specifying Statistical Models: From Parametric to Non-Parametric, Using Bayesian or Non Bayesian Approaches. Edited by J. P. Florens, M. Mouchart, J. P. Raoult, L. Simar, and A. F. M. Smith. XI, 204 pages, 1983. Vol. 17: I. V. Basawa and D. J. Scott, Asymptotic Optimal Inference for Non-Ergodic Models. IX, 170 pages, 1983. Vol. 18: W. Britton, Conjugate Duality and the Exponential Fourier Spectrum. V, 226 pages, 1983. Vol. 19: L. Fernholz, von Mises Calculus For Statistical Functionals. VIII, 124 pages, 1983. Vol. 20: Mathematical Learning Models - Theory and Algorithms: Proceedings of a Conference. Edited by U. Herkenrath, D. Kalin, W. Vogel. XIV, 226 pages, 1983. Vol. 21: H. Tong, Threshold Models in Non-linear Time Series Analysis. X, 323 pages, 1983. Vol. 22: S. Johansen, Functional Relations, Random Coefficients and Nonlinear Regression with Application to Kinetic Data. VIII, 126 pages. 1984. Vol. 23: D. G. Saphire, Estimation of Victimization Prevalence Using Data from the National Crime Survey. V, 165 pages. 1984. Vol. 24: T. S. Rao, M. M. Gabr, An Introduction to Bispectral Analysis and Bilinear Time Series Models. VIII, 280 pages, 1984. Vol. 25: Time Series Analysis of Irregularly Observed Data. Proceedings, 1983. Edited by E. Parzen. VII, 363 pages, 1984. Lecture Notes in Statistics Edited by J. Berger, S. Fienberg, J. Gani, and K. Krickeberg 45 J.K. Ghosh Editor Statistical Information and Likelihood A Collection of Critical Essays by Dr. D. 8asu Springer-Verlag New York Berlin Heidelberg London Paris Tokyo 1.K. Ghosh Indian Statistical Institute Calcutta 700 035 India Mathematics Subject Classification (1980): 62AlO Library of Congress Cataloging-in-Publication Data Ghosh,1.K Statistical information and likelihOod (Lecture notes in statistics; 45) Bibliography: p. 1. Estimation theory. 2. Basu, D. (Dev) I. Ghosh, 1.K II. Title. III. Series: Lecture notes in statistics (Springer-Verlag) ; v. 45. QA276.8.B393 1988 519.5'44 88-4931 © 1988 by Springer-Verlag New York Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag, 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or here after developed is forbidden. The use of general descriptive names, trade names, trademarks, etc. in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Mer chandise Marks Act, may accordingly be used freely by anyone. Camera-ready text prepared by the editor. 98765 4 3 2 1 ISBN -13:978-0-387-96751-6 e-ISBN-13: 978-1-4612-3894-2 DOl: 10.1007/978-1-4612-3894-2 DEDICATED To the Fond and ColoUlful Memories of SIR RONALD AYLMER FISHER (1890-1962) and PROFESSOR JERZY NEYMAN (1894-1981) FOREWORD It is an honor to be asked to write a foreword to this book, for I believe that it and other books to follow will eventually lead to a dramatic change in the current statistics curriculum in our universities. I spent the 1975-76 academic year at Florida State University in Tallahassee. My purpose was to complete a book on Statistical Reliability Theory with Frank Proschan. At the time, I was working on total time on test processes. At the same time, I started attending lectures by Dev Basu on statistical inference. It was Lehmann's hypothesis testing course and Lehmann's book was the text. However, I noticed something strange - Basu never opened the book. He was obviously not following it. Instead, he was giving a very elegant, measure theoretic treatment of the concepts of sufficiency, ancillarity, and invariance. He was interested in the concept of information - what it meant. - how it fitted in with contemporary statistics. As he looked at the fundamental ideas, the logic behind their use seemed to evaporate. I was shocked. I didn't like priors. I didn't like Bayesian statistics. But after the smoke had cleared, that was all that was left. Basu loves counterexamples. He is like an art critic in the field of statistical inference. He would find a counterexample to the Bayesian approach if he could. So far, he has failed in this respect. In 1979, Basu wrote the following : "It is about 12 years now that I finally came to the sad conclusion that most of the statistical methods that I had learned from pioneers like Karl Pearson, Ronald Fisher and Jerzy Neyman and survey practi tioners like Morris Hanson, P. C. Mahalanobis and Frank Yates are logically unten able." I believe he is right. Read Basu. Richard Barlow PREFACE These essays on foundations of Statistical Inference have been brought together in a single volume to honour Dr Basu, as he is affectionately known to his colleagues in the Indian Statistical Institute. The essays are among the most significant con tributions of our time to questions of foundation. The sequence in which they have been arranged, in consultation with Dr. Basu, makes it possible to read them (at least Parts I and II) as a single contemporary discourse on the likelihood principle, the paradoxes that attend its violation and the radical deviation from classical statis tical practices that its adoption would entail. One may also read them, with the aid of the author's notes, as a record of a personal quest. If one accepts the likelihood principle, one is led almost inevitably to a Baye sian pO:3ition. Basu is a Bayesian too. However, he holds this view in a tentative rather than dogmatic way, even though his rejection of almost all classical statistics is complete. This tentative Bayesian view appears in several chapters but specially ill chapters VII an::! XVII where it is evident in the position that he adopts regarding elimination of nuisance parameters or partial likelihood. It also appears in his reluc tance, expressed in many private discussions, to put a prior on a large or infinite dimensional parameter space. The relation of Part III to the earlier parts needs to be made clear. The first three chapters are a rounding off of some of the earlier discussions. Three, namely chapters XXII, XXIII and XXIV present a few well-known results of Basu which appear directly or indirectly in many of the examples in Part I. Chapters XX and XXI are on an entirely different aspect of foundations. They examine the impact of choice of loss functions on the notion of efficient estimates in small and large samples. Chapter XX contains a beauti ful construction which shows the existence of a best unbiased estimate depends crucially on convexity of the loss function. Chapter XXI contains the germ of a seminal idea which took its final shape in the hands of Baha duro Both are forerunners of concern about robustness under varying loss functions. The following biographical details came out in course of an interview with him in October 1987. Basu had his first course of Statistics from Professor M.C. Chakrabarty in the honours programme in Mathematics at Dacca University in what is now Bangla desh. This was around 1945. He got his master's degree in Mathematics also from Dacca and taught there from 1947 to 1948 when he moved to Calcutta. This was the time of independence, and partition of India when the whole subcontinent was in turmoil. Basu spent some time in an insurance company, trying to become an actu ary. Realising that this was not his vocation, he returned to Dacca as a Senior Preface Lecturer in 1950. After a few weeks he came to Calcutta again and joined the Indian Statistical Institute as a research scholar under Professor C. R. Rao. After submitting his thesis in 1953 he went to Berkeley as a Fullbright scholar. He came back a cO'Tlplete Neyman-Pearsonian. His first doubts began to form when he learnt about ancillaries and conditional inference fro'Tl Fisher during Fisher's visit to the Institute in 1955. Fro'Tl then onwards he began examining very critically both the Neyman-Pearsonian and the Fisherian framework, his examination eventually forcing him to a Bayesian point of view, via the likelihood route, which had been opened up but then abandon9d by Barnard and BirnbauTl. His final conversion, if one may call it so, took place in a rather amusing W3Y in 1968. Though not a Bayesian yet, he had been invited by Professor H. K. Nandi to be the first speaker in a Bayesian symposium at the Indian Science Congress held in Benaras in January, 1968. Basu says his lecture convinced none but himself; in course of preparina for that lecture he became a Bayesian. The polemical papers on foundation began to appear after 1968. The papers are being reproduced more or less as they were originally published, except for the fact that occasionally a long paper has been broken up into more than one chapter and the reference style has been made uniform. A few inconsistencies in spellings and references remain. References to earlier chapters are sometimes in the form of references to the corresponding papers or parts thereof. Moreover, all references to contemporary or past events remain as they originally appeared. Bringing out this volume would not have been possible without the help and co-operation of many people. Professor Barlow was kind enough to write a foreword. Dr. Basu has provided a summing up and author's notes at my request. I would like to thank all my colleagues in the Institute who spent a lot of time looking through the manuscript but a special mention must be made of Tapas Samanta and Sumitra Purkayastha who put in an enormous amount of work. My thanks are due to Subhas Dutta for typing the manuscript and to Prandhan Nandy for taking care of all the correspondence related to the publication of the monograph and much else. Indian Statistical Institute J. K. Ghosh Calcutta 700 035 November, 1987 viii ACKNOWLEDGEMENT I am grateful to Professor G. A. Barnard Professor O. Barndorff-Nielsen Professor D. R. Cox Professor A. P. Dempster Professor A. W. F. Edwards FJro fessor V. P. Godambe Professor David V. Hinkley Professor J.D. Kalbfleisch Professor Oscar Kempthorne Professor J. C. Koop Professor David A. Lane Pro fessor S. L. Lauri tzen Professor J. N. K. Rao Professor R. Royall Professor D. B. Rubin for their permission to include their discussion on Dr. Basu's papers and to Academic Press American Statistical Association Concordia University Hall, Rinehart a!ld Winston Insti tute 0 f Mathematical Statistics Interna tiona I Statistical Insti tute John Wiley & Sons Limited North Holland Publishing Company Statistical Publishing Society University of California Press Uni versi t y 0 f North Carolina University of Sao Paulo for glvmg permission to reproduce material published by them. The exact source is indicated in the Reference List at the end of the volume. J. K. G. CONTENTS Foreword Editor's Preface Acknowledgement Author's Summing Up PART I INFORMA nON AND LIKELIHOOD Chapter I Recovery of Ancillary Information O. Notes 1. Introduction 3 2. The Sample Size Analogy 9 3. A Logical Difficulty 13 4. Conceptual Statistical Experiments 16 Chapter II Statistical Information and Likelihood Part I : Principles 20 O. Notes 20 1. Statistical Information 21 2. Basic Definitions and Relations 23 3. Some Principles of Inference 27 4. Information as a Function 31 5. Fisher Information 33 6. The Likelihood Principle 35 Chapter III Statistical Information and Likelihood Part II : Methods 43 43 O. Notes Contents 1. Non-Bayesian Likelihood Methods 45 2. Likelihood: Point Function or a Measure? 48 3. Maximum Likelihood 55 Chapter IV Statistical Information and Likelihood Part III : Paradoxes 60 O. Notes 60 1. A Fallacy of Five Terms 61 2. The Stopping Rule Paradox 64 3. The Stein Paradox 70 Chapter V Statistical Information and Likelihood : Discussions 78 O. Notes 78 1. Discussions 79 2. Barnard-Basu Correspondence 88 Chapter VI Partial Sufficiency 98 O. Notes 98 1. Introduction 99 2. Specific Sufficient Statistics 101 3. Partial Sufficiency 102 4. H-sufficiency 104 5. Invariantly Sufficient Statistics 108 6. Final Remarks 111 Chapter VII Elimination of Nuisance Parameters 114 O. Notes 114 1. The Elimination Problem and Methods 115 2. Marginalization and Conditioning 117 xii
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