Statistical Decision Theory and Related Topics V Shanti S. Gupta James O. Berger Editors Statistical Decision Theory and Related Topics V With 68 Illustrations Springer-Verlag New York Berlin Heidelberg London Paris Tokyo Hong Kong Barcelona Budapest Shanti S. Gupta James O. Berger Department of Statistics Purdue University West Lafayette, IN 47907 USA Library of Congress Cataloging-in-Publication Data Statistical decision theory and related topics V IShanti S. Gupta, James O. Berger, editors. p. cm. Papers of the Fifth Purdue International Symposium on Statistical Decision Theory and Related Topics held June 14-19, 1992, at Purdue University Includes bibliographical references. ISBN-13:978-l-4612-7609-8 e-ISBN-13:978-l-4612-26l8-5 DOl: 10.1 007/978-1-4612-2618-5 1. Statistical decision-Congresses. I. Gupta, Shanti Swarup, 1925- . II. Berger, James O. III. Purdue International Symposium on Statistical Decision Theory and Related Topics (5th: 1992: Purdue University) QA279.4.S745 1994 519.5' 42-dc20 93-27603 Mathematics Subject Classification (1991); 62C05, 62-06 Printed on acid-free paper. © 1994 Springer-Verlag New York, Inc. Softcover reprint of the hardcover 1st edition 1994 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 New York, Inc., 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 hereaf­ ter 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 Merchandise Marks Act, may accordingly be used freely by anyone. Production managed by Francine McNeill; manufacturing supervised by Vincent Scelta. Camera-ready copy prepared by the editors using LaTeX. 987654321 PREFACE The Fifth Purdue International Symposium on Statistical Decision The­ ory and Related Topics was held at Purdue University during the period of June 14-19,1992. The symposium brought together many prominent leaders and younger researchers in statistical decision theory and related areas. The format of the Fifth Symposium was different from the previous symposia in that in addition to the 54 invited papers, there were 81 papers presented in contributed paper sessions. Of the 54 invited papers presented at the sym­ posium, 42 are collected in this volume. The papers are grouped into a total of six parts: Part 1 - Retrospective on Wald's Decision Theory and Sequential Analysis; Part 2 - Asymptotics and Nonparametrics; Part 3 - Bayesian Analysis; Part 4 - Decision Theory and Selection Procedures; Part 5 - Probability and Probabilistic Structures; and Part 6 - Sequential, Adaptive, and Filtering Problems. While many of the papers in the volume give the latest theoretical developments in these areas, a large number are either applied or creative review papers. We are extremely grateful to the administration of Purdue University, and to the School of Science, for encouragement and financial support of the symposium. The symposium was also supported by the Air Force Of­ fice of Scientific Research, by the National Science Foundation under Grant Number DMS-9118808, by the Army Research Office under Grant Num­ ber DAAL03-92-G-0029, and by the National Security Agency under Grant Number MDA904-92-H-4007. We sincerely thank these agencies for their assistance, and in particular wish to thank Dr. Deborah Lockhart of the National Science Foundation, Dr. Jon A. Sjogren of the Air Force Office of Scientific Research, Dr. Gerald R. Andersen of the Army Research Office, and Dr. Charles Osgood of the National Security Agency. Many individuals contributed to the success of the Fifth Purdue Sympo­ sium. The program for the symposium was developed under the guidance of a Scientific Organizing Committee composed of Lawrence D. Brown, Cornell University, Jayanta K. Ghosh, Indian Statistical Institute and Purdue Uni­ versity, lain Johnstone, Stanford University, Carl Morris, Harvard University, Willem R. van Zwet, Leiden University, and James V. Zidek, University of British Columbia. We two acted as the co-chairmen of this committee. The excellence of the program was due in large part to the efforts of these col­ leagues. Numerous faculty, staff, and students were instrumental in seeing that lo­ cal arrangements went smoothly. Norma Lucas provided invaluable assistance in the local organization as did Teena Seele. v VI PREFACE Colleagues at Purdue and elsewhere also graciously served as referees for the papers appearing in this volume. We thank them for their efforts, which in many cases resulted in a strengthening of the papers. Finally, the superb appearance of the volume is due to the extensive efforts of Norma Lucas and Teena Seele, who not only did a masterful job of typing many of the papers, but also performed the arduous task of achieving a uniform format and appearance. Mark Senn's help was also invaluable in this regard. We are extremely grateful to all these individuals. Shanti S. Gupta James o. Berger CONTENTS PREFACE v CONTRIBUTORS xi PART 1: RETROSPECTIVE ON WALD'S DECISION THEORY AND SEQUENTIAL ANALYSIS Minimaxity, More or Less 1 Lawrence D. Brown A Retrospective of Wald's Sequential Analysis - Its Relation 19 to Change-point Detection and Sequential Clinical Trials David Siegmund Group Bayes Estimation of the Exponential Mean: ARetrospective 35 View of the Wald Theory Constance van Eeden and James V. Zidek PART 2: ASYMPTOTICS AND NONPARAMETRICS Ibragimov - Hasminskii Models 51 P. J. Bickel and Y. Ritov Penalized Likelihood Hazard Estimation: Algorithm and Examples 61 Chong Gu Estimating Lifetime Distribution Under Different Sampling Plans 73 S. Y. He and Grace L. Yang Estimating the Mixing Density of a Mixture of Power Series 87 Distributions Wei-Liem Loh Estimating Linear Functionals in Density Estimation Models 99 Mark G. Low On Robust Estimation In Case-Control Studies with Errors in Covariates 107 C. Y. Wang and R. J. Carroll PART 3: BAYESIAN ANALYSIS Applications and Limitations of Robust Bayesian Bounds and Type II MLE 121 M. J. Bayarri and James O. Berger Bayesian Estimation of Political Transition Matrices 135 Jose M. Bernardo Using Historical Controls in Clinical Trials: Application to ECMO 141 Donald A. Berry and Janis P. Hardwick vii viii CONTENTS A Geometric Solution of the Bayesian E-Optimal Design Problem 157 H. Delle and W. J. Studden Bayesian Estimation for Censored Data: 171 An Experiment in Sensitivity Analysis Hani Doss Stability and Convergence of the Posterior in Non-Regular Problems 183 Jayanta K. Ghosh, Subhashis Ghosal and Tapas Samanta A Hierarchical Bayesian Analysis of Circular Data with Autoregressive 201 Errors: Modeling the Mechanical Properties of Cortical Bone M. Peruggia, T. J. Santner, Y- Y. Ho and N. J. McMillan Discovery Sampling and Selection Models 221 Mike West PART 4: DECISION THEORY AND SELECTION PROCEDURES Loss Functions for Set Estimation 237 George Casella, J. T. Gene Hwang and Christian P. Robert Double Sample Estimation When Cost Depends on the Parameter 253 Arthur Cohen and H. B. Sackrowitz On Some Bayesian Solutions of the Neyman-Scott Problem 267 Malay Ghosh Empirical Bayes Two-Stage Procedures for Selecting the Best 277 Bernoulli Population Compared with a Control Shanti S. Gupta, TaChen Liang and Re-Bin Rau On the Differentiation of Certain Probabilities with Applications in 293 Statistical Decision Theory L. R. HaJJ and John E. Alcaraz Minimax Bayes, Asymptotic Minimax and Sparse Wavelet Priors 303 lain M. Johnstone Simulating the Power of Decision Rules Applied to Data From the 327 HGPRT Gene Mutation Assay J. Lehn and S. Rettig Analyzing Randomized Response Data with a Binomial 341 Selection Procedure Gary C. McDonald Improved Confidence Statements for the Usual Multivariate Normal 351 Confidence Set Christian Robert and George Casella Improving on Truncated Estimators 369 Peter Yi-Shi Shao and William E. Strawderman Bayesian and Non-Bayesian Estimation Using Balanced Loss Functions 377 Arnold Zellner CONTENTS ix PART 5: PROBABILITY AND PROBABILISTIC STRUCTURES Distributions Which Are Gaussian Convolutions 391 Anirban DasGupta An Infinite Dimensional Convolution Theorem 401 L. Le Cam How Many Geometric (P) Samples Does It Take to See All the 413 Balls in a Box Thomas Sellke and John Overtieck The Asymptotic Distribution of Point Charges on a Conducting Sphere 427 Willem R. van Zwet Permutation Invariant Upper and Lower Probabilities 431 Larry Wasserman and Joseph B. Kadane Subgraph Approximations for Directed Graphical Models 441 Constantin T. Yiannoutsos and Alan E. Gelfand Lower Bounds on Expected Redundancy for Classes of Continuous 453 Markov Sources Bin Yu PART 6: SEQUENTIAL, ADAPTIVE, AND FILTERING PROBLEMS Random Walks for Quantile Estimation 467 Stephen D. Durham and Nancy Flournoy Some Procedures for State Estimation of a Hidden Markov Chain 477 with Two States R. Khasminskii, B. Lazareva and J. Stapleton Adaptive Estimation Via Martingales 489 Tze Leung Lai The Blackwell Prediction Algorithm for Infinite 0-1 Sequences, 503 and a Generalization H. R. Lerche and J. Sarkar Some Statistical Problems in Multitarget Tracking 513 C. Radhakrishna Rao A Generalized Parking Problem 523 Michael Woodroofe, Rudi Lerche and Robert Keener Search Methods and Observer Logics 533 H. P. Wynn, A. A. Zhigljavsky and J. H. 0 'Geron CONTRIBUTORS John E. Alcaraz, Department of Mathematics, 0112, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0112 M. J. Bayarri, Departamento de Estadistica, University of Valencia, C/NAVE 2, 46001 Valencia, Spain James O. Berger, Department of Statistics, Purdue University, 1399 Mathematical Sciences Building, West Lafayette, IN 47907-1399 Jose M. Bernardo, Departamento de Estadistica, Presidencia de la Generalitat, Ca­ balleros 2, 46001 Valencia, Spain Donald A.