Parametric statistical models and likelihood
type: Item
instance of: scholarly article
title: Parametric statistical models and likelihood
zbMATH Open document ID: 0691.62002
published in: Lecture Notes in Statistics
publication date: 1993-06-05T00:00:00Z
review text: According to the author these lectures notes concern some mathematically advanced aspects of parametric statistical inference. There is an introductory chapter (Chapter 0) which outlines the contents of the remaining seven chapters. Chapter 1 is devoted to the discussion of one of the central concepts of parametric inference, viz likelihood, and its relation to various other ideas such as sufficiency and ancillarity. Topics discussed in this chapter include parametrization invariance, marginal and conditional likelihood, pseudo likelihood, profile likelihood and modified profile likelihood, partial sufficiency and partial ancillarity and key expansions for various likelihood quantities. The two main general classes of parametric models, transformation models and exponential models and their structures are discussed in Chapter 2. As a preliminary step, some basic aspects of the theory of group actions and invariant measures are outlined. Chapter 3 may be considered a self-contained introduction to differential geometry. The concepts discussed in this chapter include multiarrays, tensors and affine connections, strings, covariant differentiation, intertwining, geometric measures, manifolds with Lie group actions, fibre bundles, connections and transport. The geometry of parametric models is discussed in Chapter 4. This chapter also discusses construction of approximately ancillary statistics. Chapter 5 is devoted to a discussion of cumulants of m-dimensional random variables and the relations between moments and cumulants. Chapter 6 is concerned with asymptotic expansions including Laplace's method, Edgeworth expansions and saddle-point approximations. Chapter 7, which is the final chapter, draws on all previous chapters. The distributional properties of various likelihood quantities are discussed here. These include the maximum likelihood estimator, the score vector and likelihood ratio statistics. The notion of modified profile likelihood is also treated in this cha…
Mathematics Subject Classification ID: 62-02
Mathematics Subject Classification ID: 62E10
Mathematics Subject Classification ID: 62F99
Mathematics Subject Classification ID: 62A01
Mathematics Subject Classification ID: 62B05
Mathematics Subject Classification ID: 62F12
Mathematics Subject Classification ID: 62E20
Mathematics Subject Classification ID: 62F05
Mathematics Subject Classification ID: 62F10
Mathematics Subject Classification ID: 62B99
Mathematics Subject Classification ID: 53A45
Mathematics Subject Classification ID: 53A99
Mathematics Subject Classification ID: 53B05
Mathematics Subject Classification ID: 53C99
zbMATH DE Number: 194591
zbMATH Keywords: parametrization invariance
zbMATH Keywords: conditional likelihood
zbMATH Keywords: pseudo likelihood
zbMATH Keywords: profile likelihood
zbMATH Keywords: modified profile likelihood
zbMATH Keywords: partial sufficiency
zbMATH Keywords: partial ancillarity
zbMATH Keywords: transformation models
zbMATH Keywords: exponential models
zbMATH Keywords: group actions
zbMATH Keywords: invariant measures
zbMATH Keywords: multiarrays
zbMATH Keywords: tensors
zbMATH Keywords: strings
zbMATH Keywords: covariant differentiation
zbMATH Keywords: intertwining
zbMATH Keywords: geometric measures
zbMATH Keywords: fibre bundles
zbMATH Keywords: transport
zbMATH Keywords: geometry of parametric models
zbMATH Keywords: approximately ancillary statistics
zbMATH Keywords: cumulants of m-dimensional random variables
zbMATH Keywords: moments
zbMATH Keywords: asymptotic expansions
zbMATH Keywords: Laplace's method
zbMATH Keywords: Edgeworth expansions
zbMATH Keywords: saddle- point approximations
zbMATH Keywords: maximum likelihood estimator
zbMATH Keywords: score vector
zbMATH Keywords: likelihood ratio statistics
author: Ole E. Barndorff-Nielsen
reviewed by: Hariharan K. Iyer
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 194591