## Doctoral Theses

### Sufficiency Pairwise Sufficiency and Bayes Sufficiency in Undominanted Experiments.

2-28-1980

2-28-1981

#### Institute Name (Publisher)

Indian Statistical Institute

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

Mathematics

#### Department

Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)

#### Supervisor

Ghosh, Jayanta Kumar (TSMU-Kolkata; ISI)

#### Abstract (Summary of the Work)

An experiment or a statiatical structure consists of a set X and a family of probability measures P on , indexod by a set (. The set I together with a g-algebra of subsets of X is the sample space and the set (H ith a -algebra g of subsets of (D is the parameter space, To avoid trivialities ve consider only situations where the para- sotrization is one-one, i,e, if Ã˜1, and Ã˜2, are distinct then s0 are PÃ˜1. and PÃ˜2. Various notions of aufficiency of a -lgebra 3 is considered in statistical literature, Anong the se are the following.(1) claasical. Conlitional probability of (X, 4) si ven E 1a indepondent of Ã˜.(ii) Decision Theoretic. For every decision problem given any decision procedure there is an equi valent decision procedure basod on b.(iii) Bayes. Given any prior on (H), the posterior distribution of e given (x,A ).is the Bame aa the posterior diatribution given (x, B). In what follows we will refer to (i) simply as sufficiency (ii) and (iii) vill be called D-sufficiency and Bayea sufficiene respectively.(i), (ii) and (iii) are known to be equivalent when (H) is dominated by a a-finite measure. Durlholder's example of a non-sufficient a-algebra containing a sufficient 0-algebra shows that in the und ominated case neither (ii) nor (iii) is equivalent to (i), In this thesis we inve stigate the relationship between (i), (ii) and (iii) when all the a-algebres involved are countably generated, Interest in countably gener- a ted o-algebras stems from the fact that those and only tho so arise out of real valued functions,Our attempts center around a conjecture of Blaclovell. During a conversation, in the winter of 77, Blackwell conjec- tured that, when both the sample space and the parameter apace are standard Borel, then for countably generated o-algebras (i), (ii) and (iii) would be equivalent, (i) and (ii) tun out to be equivalent without any Standard Borel assumptions, However the situation is different in case of (i) and (iii), Examples show that without Standard Borel assumptions (iii) noed not show, imply (i). We can/ and do so in this thesis, that for a class of Standard Borel experimonts (iii) and (i) are indeod equi- valont, The general case still remains unsolved.In chapter I, we atudy the first part of Blackwoll's conjecture, vis, (i) (ii), It is shown there that a OF compreon COR b optimtlon copy of CVIBION POFOemp - 3 D-sufficiont o-algebra always contains a sufficient o-algobra. A theorem of Burkholder then establishes the oquivalence of (1) and (ii) in the countably gonera ted case.Chapter II is dovoted to a st udy of Bayes sufficioney in the countably generated caso, The first the orem relatee nayes sufficiency to sufficiency on sets of measuro 1, More procisely a o-algebra E is Bayes sufficient iff for every

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