The theory and practice of decision making involves infinite or finite number of actions. The decision rules with a finite number of elements in the action space are the so-called multiple decision procedures. Several approaches to problems of multi ple decisions have been developed; in particular, the last decade has witnessed a phenomenal growth of this field. An important aspect of the recent contributions is the attempt by several authors to formalize these problems more in the framework of general decision theory. In this work, we have applied general decision theory to develop some modified principles which are reasonable for problems in this field. Our comments and contributions have been written in a positive spirt and, hopefully, these will an impact on the future direction of research in this field. Using the various viewpoints and frameworks, we have emphasized recent developments in the theory of selection and ranking ~Ihich, in our opinion, provides one of the main tools in this field. The growth of the theory of selection and ranking has kept apace with great vigor as is evidenced by the publication of two recent books, one by Gibbons, Olkin and Sobel (1977), and the other by Gupta and Panchapakesan (1979). An earlier monograph by Bechhofer, Kiefer and Sobel (1968) had also provided some very interest ing work in this field.

Shanti S. Gupta has made pioneering contributions to ranking and selection theory; in particular, to subset selection theory. His list of publications and the numerous citations his publications have received over the last forty years will amply testify to this fact. Besides ranking and selection, his interests include order statistics and reliability theory. The first editor's association with Shanti Gupta goes back to 1965 when he came to Purdue to do his Ph.D. He has the good fortune of being a student, a colleague and a long-standing collaborator of Shanti Gupta. The second editor's association with Shanti Gupta began in 1978 when he started his research in the area of order statistics. During the past twenty years, he has collaborated with Shanti Gupta on several publications. We both feel that our lives have been enriched by our association with him. He has indeed been a friend, philosopher and guide to us.

The theory and practice of decision making involves infinite or finite number of actions. The decision rules with a finite number of elements in the action space are the so-called multiple decision procedures. Several approaches to problems of multi ple decisions have been developed; in particular, the last decade has witnessed a phenomenal growth of this field. An important aspect of the recent contributions is the attempt by several authors to formalize these problems more in the framework of general decision theory. In this work, we have applied general decision theory to develop some modified principles which are reasonable for problems in this field. Our comments and contributions have been written in a positive spirt and, hopefully, these will an impact on the future direction of research in this field. Using the various viewpoints and frameworks, we have emphasized recent developments in the theory of selection and ranking ~Ihich, in our opinion, provides one of the main tools in this field. The growth of the theory of selection and ranking has kept apace with great vigor as is evidenced by the publication of two recent books, one by Gibbons, Olkin and Sobel (1977), and the other by Gupta and Panchapakesan (1979). An earlier monograph by Bechhofer, Kiefer and Sobel (1968) had also provided some very interest ing work in this field.

Some auxiliary results: monotonicity properties of probability distributions; Multiple decision theory: a general approach; Modified minimax decision procedures; Invariant decision procedures; Robust selection procedures: most economical multiple decision rules; Multiple decision procedures based on tests.

Decision theory is generally taught in one of two very different ways. When of opti taught by theoretical statisticians, it tends to be presented as a set of mathematical techniques mality principles, together with a collection of various statistical procedures. When useful in establishing the optimality taught by applied decision theorists, it is usually a course in Bayesian analysis, showing how this one decision principle can be applied in various practical situations. The original goal I had in writing this book was to find some middle ground. I wanted a book which discussed the more theoretical ideas and techniques of decision theory, but in a manner that was constantly oriented towards solving statistical problems. In particular, it seemed crucial to include a discussion of when and why the various decision prin ciples should be used, and indeed why decision theory is needed at all. This original goal seemed indicated by my philosophical position at the time, which can best be described as basically neutral. I felt that no one approach to decision theory (or statistics) was clearly superior to the others, and so planned a rather low key and impartial presentation of the competing ideas. In the course of writing the book, however, I turned into a rabid Bayesian. There was no single cause for this conversion; just a gradual realization that things seemed to ultimately make sense only when looked at from the Bayesian viewpoint.

This book is a slightly revised and expanded version of a set I I I of notes used for a lecture series given at the Ecole dlEte de I Probabilites at st. Flour in August 1986. In view of the statistical nature of the material discussed herein it was agreed to publish the material as a separate volume in the statistics series rather than, as is the tradition, in a joint volume in the Lecture Notes in Mathematics Series. It is a genuine pleasure to have this opportunity to thank I I I the organizers of Les Ecoles dlEte, and in particular Professor P. -L. Hennequin, for the excellent arrangements of these Summer Schools which form a very significant forum for the exchange of scientific ideas relating to probability. The efficient, careful and patient preparation of the typescript by Oddbj~rg Wethelund is also gratefully acknowledged. Aarhus, June 1988 O. E. Barndorff-Nielsen Parametric statistical Models and Likelihood O. E. Barndorff-Nielsen o. Introduction 0. 1. Outline of contents 1 0. 2. A few preliminaries 2 1. Likelihood and auxiliary statistics 1. 1. Likelihood 4 1. 2. Moments and cumulants of log likelihood derivatives 10 1. 3. Parametrization invariance 13 1. 4. Marginal and conditional likelihood 15 * 1. 5. Combinants, auxiliaries, and the p -model 19 1. 6. Orthogonal parameters 27 1. 7. Pseudo likelihood, profile likelihood and modified 30 profile likelihood 1. 8. Ancillarity and conditionality 33 41 1. 9. Partial sufficiency and partial ancillarity 1. 10.

The concept of conditional specification is not new. It is likely that earlier investigators in this area were deterred by computational difficulties encountered in the analysis of data following con ditionally specified models. Readily available computing power has swept away that roadblock. A broad spectrum of new flexible models may now be added to the researcher's tool box. This mono graph provides a preliminary guide to these models. Further development of inferential techniques, especially those involving concomitant variables, is clearly called for. We are grateful for invaluable assistance in the preparation of this monograph. In Riverside, Carole Arnold made needed changes in grammer and punctuation and Peggy Franklin miraculously transformed minute hieroglyphics into immaculate typescript. In Santander, Agustin Manrique ex pertly transformed rough sketches into clear diagrams. Finally, we thank the University of Cantabria for financial support which made possible Barry C. Arnold's enjoyable and productive visit to S- tander during the initial stages of the project. Barry C. Arnold Riverside, California USA Enrique Castillo Jose Maria Sarabia Santander, Cantabria Spain January, 1991 Contents 1 Conditional Specification 1 1.1 Why? ............. ........ . 1 1.2 How may one specify a bivariate distribution? 2 1.3 Early work on conditional specification 4 1.4 Organization of this monograph . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 5 2 Basic Theorems 7 Compatible conditionals: The finite discrete case.