Extremal Families and Systems of Sufficient Statistics

The pOint of view behind the present work is that the connection between a statistical model and a statistical analysis-is a dua­ lity (in a vague sense). In usual textbooks on mathematical statistics it is often so that the statistical model is given in advance and then various in­ ference principles are applied to deduce the statistical ana­ lysis to be performed. It is however possible to reverse the above procedure: given that one wants to perform a certain statistical analysis, how can this be expressed in terms of a statistical model? In that sense we think of the statistical analysis and the stati­ stical model as two ways of expressing the same phenomenon, rather than thinking of the model as representing an idealisation of "truth" and the statistical analysis as a method of revealing that truth to the scientist. It is not the aim of the present work to solve the problem of giving the correct-anq final mathematical description of the quite complicated relation between model andanalysis. We have rather restricted ourselves to describe a particular aspect of this, formulate it in mathematical terms, and then tried to make a rigorous and consequent investigation of that mathematical struc­ ture.

I The Case of a Single Experiment and Finite Sample Space.- 1. Basic facts. Maximal and extremal families.- 2. Induced maximal and extremal families.- 3. Convexity, maximal and extremal families.- 4. Some examples.- II Simple Repetitive Structures of Product Type. Discrete Sample Spaces.- 0. Conditional independence.- 1. Preliminaries. Notation.- 2. Notions of sufficiency.- 3. Maximal and extremal families.- 4. Limit theorems for maximal and extremal families.- 5. The topology of $$\left( {\mathop{{\dot{U}}}\limits_{n} {{y}_{n}}} \right)UM. $$ Boltzmann laws.- 6. Integral representation of M.- 7. Construction of maximal and extremal families.- 8. On the triviality of the tail ?-algebra of a Markov chain.- 9. Examples of extremal families.- 10. Bibliographical notes.- III Repetitive Structures of Power Type. Discrete Sample Spaces.- 0. Basic facts about Abelian semigroups.- 1. Extremal families for semigroup statistics.- 2. General exponential families.- 3. The classical case.zd-valuedstatistics.- 4. Maximum likelihood estimation in general exponential families.- 5. Examples of general exponential families.- 6. Bibliographical notes.- IV General Repetitive Structures of Polish Spaces. Projective Statistical Fields.- 0. Probability measures on Polish spaces.- 1. Projective systems of Polish spaces and Markov kernels.- 2. Projective statistical fields.- 3. Canonical projective statistical fields on repetitive structures.- 4. Limit theorems for maximal and extremal families on repetitive structures.- 5. Poisson Models.- 6. Exponential Families.- 7. Examples from continuous time stochastic processes.- 8. Linear normal models.- 9. The Rasch model for item analysis.- 10. Bibliographical notes.- Literature.