Extremal Families and Systems of Sufficient Statistics

ISBN-13: 9780387968728 / Angielski / Miękka / 1988 / 268 str.

Steffen L. Lauritzen

Opis

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 and analysis. 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."

Spis treści

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 $$\left( {\mathop{{\dot}}\limits_ } \right)UM. $$ 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-valued statistics.- 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.

O autorze

Steffen L. Lauritzen is Professor of Statistics and Head of the Department of Statistics at the University of Oxford.