In the present work I want to show a mathematical study of the statistical notion of sufficiency mainly for undominated statistical experiments. The famous Burkholder's (1961) and Pitcher's(1957) examples motivated some researchers to develop new theory of sufficiency. Le Cam (1964) is probably the most excellent paper in this field of study. This note also belongs to the same area. Though it is more restrictive than Le Cam's paper(1964), a study which is connected more directly with the classical papers of Halmos and Savage(1949) , and Bahadur(1954) is shown. Namely I want to develop a study based on the notion of pivotal measure which was introduced by Halmos and Savage(1949) . It is great pleasure to have this opportunity to thank Professor H. Heyer and Professor H. Morimoto for their careful reading the manuscript and valuable comments on it. I am also thankful to Professor H. Luschgy and Professor D. Mussmann for thei r proposal of wr i ting "the note". I would like to dedicate this note to the memory of my father Eizo.

Klappentext

This monograph presents an approach to the measure-theoretical foundations of statistics and the theory of sufficiency, covering undominated and dominated statistical experiments. The familiar topics in the dominated case, such as pairwise sufficiency, Neyman factorization, minimal sufficient statistics, and the Rao-Blackwell theorem, are treated from a more general viewpoint than in the Halmos-Savage-Bahadur scheme and sometimes in a slightly different way. The main theme is that if the usual notion of sufficiency in terms of conditional probabilities is modified to omit the standard assumption of a common dominating sigma-finite measure, then certain aspects of statistics become more straightforward. In particular, one can extend Neyman's factorization criterion. Consequently, this will be of interest to researchers in statistics and may even lead to new developments in the theory of sufficiency.

Inhalt
0 Introduction.- 1 Undominated experiments.- 1.1. Majorized experiments and their decomposition.- 1.2. Weakly dominated experiments.- 1.3. Examples.- 1.4. Bibliographical notes.- 2 PSS, pivotal measure and Neyman factorization.- 2.1. PSS and pivotal measure for majorized experiments.- 2.2. Generalizations of the Neyman factorization theorem.- 2.3. Neyman factorization and pivotal measure in the case of weak domination.- 2.4. Dominated case.- 2.5. Bibliographical notes.- 3 Structure of pairwise sufficient subfield and PSS.- 3.1. Discrete experiment case.- 3.2. Majorized experiment case.- 3.3. Burkholder problem of sufficiency and completions.- 3.4. Bibliographical notes.- 4 The Rao-Blackwell theorem and UMVUE.- 4.1. Rao-Blackwell theorem for PSS in weakly dominated experiments.- 4.2. Converse of a theorem of Lehmann and Scheffé.- 4.3. Bibliographical notes.- 5 Common conditional probability for PSS and its applications.- 5.1. Deficiency.- 5.2. Representation of M-space of majorized experiment.- 5.3. Representation of the common conditional probability for PSS by (T)-integral.- 5.4. Application of the extended notion of common conditional probability for PSS subfield.- 5.5. Bibliographical notes.- 6 Structure of pivotal measure.- 6.1. Minimal L-space.- 6.2. Maximal orthogonal system and pivotal measure.- 6.3. Bibliographical notes.- References.- List of symbols.