Parametric statistical models and likelihood
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 chapter. There is also an appendix which consists of some further mathematical tools, a list of references, and a subject index. Examples are provided in every chapter to illustrate the concepts. These lecture notes are primarily intended for researches in the field of parametric inference. Although the technical details will be inaccessible for a majority of statisticians, a determined reader would learn about many of the fundamental concepts underlying parametric statistical inference.
cumulants of m-dimensional random variables
scientific article; zbMATH DE number 194591