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.

Klappentext

The book gives an account of the mathematical-statistical theory of the main classes of parametric statistical models, i.e. transformatioon models and exponential models, and of likelihood based inference. The emphasis is on recent developments - various new results are presented - and the mathematical techniques employed include parts of the theory of group actions and invariant measures, differential geometry, and asymptotic analysis. A knowledge of these techniques is not presupposed but will be helpful, as the exposition is partly quite succinct. A basic knowledge of classic parametric statistical inference is however assumed. Exactness results and high-order asymptotic results for important likelihood quantities, including maximum likelihood estimators, score vectors, (signed) likelihood ratios and (modified) profile likelihoods, are discussed. Concepts of ancillarity and sufficiency enter prominently.

Inhalt
0. Introduction.- 0.1. Outline of contents.- 0.2. A few preliminaries.- 1. Likelihood and auxiliary statistics.- 1.1. Likelihood.- 1.2. Moments and cumulants of log likelihood derivatives.- 1.3. Parametrization invariance.- 1.4. Marginal and conditional likelihood.- 1.5. Combinants, auxiliaries, and the p*-model.- 1.6. Orthogonal parameters.- 1.7. Pseudo likelihood, profile likelihood and modified profile likelihood.- 1.8. Ancillarity and conditionality.- 1.9. Partial sufficiency and partial ancillarity.- 1.10. Likelihood expansions.- 1.11. Additional bibliographical notes.- 2. Transformation models and exponential models.- 2.1. Group actions and invariant measures.- 2.2. Transformation models.- 2.3. Transformation submodels.- 2.4. Exponential models.- 2.5. Exponential transformation models.- 2.6. Additional bibliographical notes.- 3. Reparametrizations and differential geometry.- 3.1. Multiarrays.- 3.2. Tensors and affine connections.- 3.3. Strings.- 3.4. Covariant differentiation and strings.- 3.5. Intertwining.- 3.6. Submanifolds.- 3.7. Geometric measures.- 3.8. Manifolds with a Lie group action.- 3.9. Fibre bundles, connections and (parallel) transport.- 3.10. Additional bibliographical notes.- 4. Inferential and geometric structures.- 4.1. Ancillary statistics and conditionality structures.- 4.2. Conditionality structures for transformation models.- 4.3. Construction of approximately ancillary statistics.- 4.4. Jacobians of conditionality structures.- 4.5. Geometry of parametric models.- 4.6. Additional bibliographical notes.- 5. Cumulants.- 5.1. Elemental properties of cumulants.- 5.2. Relations between moments and cumulants.- 5.3. An alternative definition of generalized cumulants.- 5.4. Additional bibliographical notes.- 6. Laplace's method. Edgeworth andsaddle-point approximations.- 6.1. Laplace's method.- 6.2. Hermite polynomials.- 6.3. Edgeworth approximations.- 6.4. Saddle-point approximations.- 6.5. Additional bibliographical notes.- 7. Distributions of likelihood quantities.- 7.1. The distribution of the maximum likelihood estimator.- 7.2. Expansion of p*.- 7.3. The distribution of the score vector.- 7.4. The distribution of likelihood ratio statistics.- 7.5. Modified profile likelihood.- 7.6. Additional bibliographical notes.- Appendices.- A.1. Taylor's formula.- A.2. Fourier transformation.- A.3. Some formulas for matrices and determinants.- A.4. Partially ordered sets, partitions and Möbius inversion.- A.5. The Legendre transform.- A.6. A differential geometric inversion result.- References.