Empirical Bayes and Likelihood Inference

241 Pages·2001·5.001 MB·English

by  T. A. Louis (auth.), S. E. Ahmed, N. Reid (eds.)

Lecture Notes in Statistics 148
Edited by P. Bickel, P. Diggle, S. Fienberg, K. Krickeberg,
I. Olkin, N. Wermuth, S. Zeger
Springer Science+Business Media, LLC
S.E. Ahmed
N. Reid (Editors)
Empirical Bayes and Likelihood Inference
Springer
S.E.Ahmed N. Reid
Department of Mathematics Department of Statistics
University of Regina University of Toronto
Regina, Saskatchewan S4S OA2 Toronto, Ontario M5S 3G3
Canada Canada
Library of Congress Cataloging-in-Publication Data
Empirical Bayes and likelihood inference / editors, S.E. Ahmed, N. Reid.
p. cm.-(Lecture notes in statistics; 148)
Inc\udes bibliographical references.
ISBN 978-0-387-95018-1 ISBN 978-1-4613-0141-7 (eBook)
DOI 10.1007/978-1-4613-0141-7
1. Bayesian statistical decision theory. 2. Estimation theory. 1. Ahmed, S.E.
(Syed Ejaz), 1957-. II. Reid, N. III. Lecture notes in statistics (Springer
Verlag); v. 148.
QA279.5.E47 2001
519.5'.42-dc21 00-059550
Printed on acid-free paper.
©2001 Springer Science+Business Media New York
Originally published by Springer-Verlag New York, Inc. in 2001
AII rights reserved. This work may not be translated or copied in whole or in part without the
written permission ofthe publisher Springer Science+Business Media, LLC 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 dis similar methodology now known or hereafter 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 be accordingly used freely by anyone.
Camera-ready copy provided by the Centre de Recherches Mathematiques.
9 8 7 6 543 2 1
ISBN 978-0-387-95018-1 SPIN 10761462
Preface
In November, 1997, the Centre de Recherches Mathematiques hosted a one
week workshop called "Empirical Bayes and Likelihood Inference" as part of
a year long program in statistics. This workshop was attended by about fifty
researchers, featured eighteen speakers and a lively discussion. In January,
1995, in the newsletter of the Royal Statistical Society, Efron predicted
that shrinkage and empirical Bayes methodology would be a major area
of statistical research for the early 21st century. Shrinkage and empirical
Bayes methods provide useful techniques for combining data from various
sources, and recent asymptotic theory has advanced understanding of the
fundamental role of the likelihood function for much the same purpose.
The goal of the workshop was to explore and create common ground in
likelihood based inference, Bayesian inference and empirical Bayes meth
ods. This last topic has until fairly recently emphasized point estimation,
while the first two are typically focussed on a distributional assessment.
The thirteen papers presented in this volume cover a range of interesting
practical problems, and illustrate some common themes as well as some
distinct differences in approach.
Empirical Bayes methods were originally developed for random effects
models, which have a natural hierarchical structure that can also be treated
by Bayesian methods. Random effects models are playing an increasingly
prominent role in applications, possibly because they provide a way to
model population heterogeneity that is easy to describe.
Several papers (Louis, Guttman, Rao, Sen) consider versions of random
effects models and aspects of inference in these models. Louis uses random
effects to model Poisson overdispersion and spatial heterogeneity, with the
''triple goals" of estimating the random effects, estimating their ranks, and
estimating their distribution. He notes that in fields such as outcomes re
search, where an end goal may well be to rank several units, it is important
to be aware of these competing goals and of the fact that the performance of
estimators can depend heavily on the goal. Guttman draws attention to the
fact that in normal theory random effects models the behaviour of posterior
point estimates depends on whether these are computed from the marginal
or joint posterior, and considers this phenomenon in relation to the EM al
gorithm. Rao uses random effects to link model-based and sampling-based
estimation methods, to address the important topic in survey sampling of
small area estimation.
vi Preface
Several papers consider the construction of point estimates and their
comparison by either Bayes risk, mean squared error, or asymptotic ver
sions of this. Sen, Ahnsanullah and Ahmed, Singh, and Ahmed all investi
gate various versions of this problem. Both Sen and Singh emphasize the
extension to (partially) nonparametric settings. Ahsanullah and Ahmed
and Ahmed study estimation in problems of interest in survival data stud
ies, the latter incorporating type II censoring.
Doksum and Normand and O'Rourke are more explicitly concerned with
applications in medical studies. Doksum and Normand consider the the
estimation of aspects of the prior distribution of an unknown changepoint,
the changepoint corresponding to infection time with the HIV virus. Accu
mulated responses on CD4 counts and covariates measured at study entry
are used to estimate the distribution of the change point. O'Rourke con
siders the relationship of empirical Bayes methods to the important area
of meta-analysis.
Estimation of the variance or mean squared error of empirical Bayes
estimators is an area needing much further study. These shrinkage type
estimators are often sufficiently complicated that this is a difficult prob
lem. In contrast, fully Bayesian and likelihood approaches typically pro
vide distributional assessments, usually asymptotic, and point estimates
with standard errors are at most a handy summary of these. Thus the
Bayesian and likelihood approach may have something to offer to the em
pirical Bayes approach, most likely by considering in more detail random
effects models, a research area of current interest. These approaches raise
various problems of their own, of course, often best addressed through de
tailed study of special cases. Yin and Ghosh investigate the construction
of so-called matching priors in in the problem of inference about a ratio
of means, and Sprott considers likelihood methods for the same problem.
Fraser and Reid show that ancillary statistics needed for higher order as
ymptotic likelihood theory can be constructed in multiparameter problems.
Hu and Zidek consider a new approach to likelihood that weights compo
nents differentially. Interestingly, this leads to shrinkage type estimators
that also feature prominently in empirical Bayes methodology.
Several directions for inference were highlighted by the talks and the
discussion, and we hope that this volume conveys some of the surprises,
puzzles and successes of theoretical statistics.
We would like to express our thanks to the superb staff at the Centre
de Recherches Mathematiques for the encouragement and support for the
organization of the theme year in statistics and this workshop in particu
lar. Marty Goldstein provided leadership and encouragement through the
planning stages for the theme year and for the production of this book,
Louis Pelletier and Josee Laferriere provided superb local arrangements for
the workshop, and Louise Letendre and Diane Poulin outstanding techni
cal support for the production of this proceedings. Funding for the theme
year was provided by CRM, NSERC (National Science and Engineering
Preface vii
Research Council of Canada), Fonds FCAR (Le Fonds pour la Formation
a
de Chercheurs et l'Aide la Recherche) of Quebec, and NATO.
S. Ejaz Ahmed
Nancy Reid
May, 2000
Contents
List of Contributors xiii
1 Bayes/EB Ranking, Histogram and Parameter Estimation:
Issues and Research Agenda 1
T.A. Louis
1 Introduction......... 1
2 Model and Inferential Goals 2
3 Triple-Goal Estimates . . . 4
4 Triple Goal Evaluations . . 5
5 Correlated O's and Unequal1k 9
6 Research Agenda 12
7 References........... 15
2 Empirical Bayes Estimators and EM Algorithms in
One-Way Analysis of Variance Situations 17
I. Guttman
1 The Model. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 17
2 Some Analysis Derived From the Hierarchical Model . . . .. 18
3 Large Sample Properties of the Estimates: The Collapsing Effect 21
4 Mean Square Considerations. . . . . . . . . . . . . . . . . .. 23
a;
5 The EM Algorithm in the One-Way Situation: Case Unknown 27
6 References............................. 32
3 EB and EBL UP in Small Area Estimation 33
J.N.K. Rao
1 Introduction. 33
2 Type 1 Model 35
3 Type 2 Model 39
4 References.. 41
4 Semiparametric Empirical Bayes Estimation in Linear
Models 45
P.K. Sen
1 Introduction............ 45
2 Preliminary Notions . . . . . . . 46
3 Empirical Bayes Interpretations . 48
4 BAN Estimators: Empirical Bayes Versions 52
5 Semiparametric Empirical Bayes Estimators 55
x Contents
4 BAN Estimators: Empirical Bayes Versions 52
5 Semiparametric Empirical Bayes Estimators 55
6 Some Concluding Remarks. 61
7 References................... 63
5 Empirical Bayes Procedures for a Change Point Problem
with Application to HIV / AIDS Data 67
S.-L. T. Normand and K. Doksum
1 Introduction........ 67
2 Gaussian Models . . . . . . . . 68
3 Estimation of Parameters . . . 71
4 Application to the SFMHS Cohort 73
5 Discussion. 77
6 References.............. 78
6 Bayes and Empirical Bayes Estimates of Survival and
Hazard Functions of a Class of Distributions 81
M. Ahsanullah and S.E. Ahmed
1 Introduction...... 81
2 Estimation Strategies. 82
3 Numerical Results 85
4 References....... 87
7 Bayes and Empirical Bayes Procedures for Selecting Good
Populations From a Translated Exponential Family 89
R.S. Singh
1 Introduction................... 89
2 Bayes and Empirical Bayes Approach 91
3 Development of Empirical Bayes Procedures . 93
4 Asymptotic Optimality of the EB Procedures and Rates of
Convergence. . . . . . . . . . . . . . . . . . . . . . 96
5 Concluding Remarks and Extension of the Results 99
6 References....................... 99
8 Shrinkage Estimation of Regression Coefficients From
Censored Data With Multiple Observations 103
S.E. Ahmed
1 Preliminaries and Introduction 103
2 Preliminary Test Estimation. . 110
3 Shrinkage Estimation. . . . . . 114
4 Positive-Part Shrinkage Estimation 116
5 Recommendations and Concluding Remarks 118
6 Appendix . 119
7 References.................... 119
Contents xi
9 Bayesian and Likelihood Inference for the Generalized
Fieller-Creasy Problem 121
M. Yin and M. Ghosh
1 Introduction....... 121
2 Likelihood Based Analysis 123
3 Noninformative Priors . . 125
4 Propriety of Posteriors . . 128
5 Simulation Study and Discussion 130
6 Appendix . 135
7 References............. 137
10 The Estimation of Ratios From Paired Data 141
D.A. Sprott
1 Introduction............................ 141
2 The Standard Analysis; Assumption (a) . . . . . . . . . . . . 143
3 An Approximate Conditional Location-Scale Model; Assump-
tion (b) ..................... 145
4 A Full Location-Scale Model; Assumption (c) 147
5 Examples .. . . . . . . . . . . . . . 150
6 The Linear Functional Relationship. 154
7 Discussion. 156
8 References............... 158
11 Meta-Analysis: Conceptual Issues of Addressing Apparent
Failure of Individual Study Replication or "Inexplicable"
Heterogeneity 161
K. O'Rourke
1 Introduction......... 161
2 MA and RCT Background. 163
3 History Overview . . . . . . 164
4 Current Likelihood Based Methods for MA 166
5 Examples of EB, CL and HB Approaches 167
6 Future Directions . . . . . 170
7 Initial Conclusions . . . . 172
8 A Bayesian Afterthought . 173
9 Final Conclusions. . . . . 174
10 Re-Analysis of Examples. 175
11 What is a MA and When Should It Be Done 179
12 References ................... . 180
12 Ancillary Information for Statistical Inference 185
D.A.;~'. Fraser and N. Reid
1 Intl'lduction............ 185
2 Third Order Statistical Inference 187
3 First Derivative Ancillary .... 189