In this and the next chapter, background material on integration, topological groups and group actions is presented. The concrete examples described below provide a direct connection between the rather abstract theory of Haar measure and its application to situations which are relevan't in statistical applications.

In this chapter, group actions are reviewed and are illustrated with examples of relevance for statistical applications. Relatively invariant integrals (measures) are defined and examples are given. An important result, due to Weil, gives necessary and sufficient conditions for the existence and uniqueness of relatively invariant integrals when the group action is transitive. A discussion of invariant and equivariant functions closes out the chapter.

In this lecture, invariant statistical models are introduced and a variety of examples is given. Invariant testing problems and equivariant estimators are introduced. Univariate and multivariate linear models provide a host of standard examples.

Our goal here is to understand the structure of probability measures which are invariant under a compact group. In the first section, a basic representation theorem is proved and is interpreted in terms of random variables. Section 2 contains some basic examples while applications to some robustness problems are given in Section 3.

The main purpose of this chapter is to discuss the extent to which Theorem 4.3 can be generalized to arbitrary Radon measures (rather than probability measures) and to cases where G is not compact. Such generalizations can be used in the derivation of densities of maximal invariants as well as in other areas. The approach here is modelled after that described in Andersson (1982). Other possible approaches to this problem are described in Wijsman (1986) (the global cross section approach using some Lie group theory) and Farrell (1985) (a measure-theoretic cross section approach developed by Schwartz (1966), unpublished).

This rather lengthy chapter provides an introduction to invariant decision problems. After describing the basic ingredients in a decision problem, invariance is introduced and used to define an invariant decision problem. A main result in this chapter shows how to construct a best invariant rule when the group action is transitive on the parameter space and when the dominating measure is decomposable [That is, the integral J defined by the measure satisfies Equation (5.14) in Theorem 5.5.] Applications of this result to the construction of best invariant estimators are given.

Orthogonal matrices, both fixed and random, play an important role in much of statistics, especially in multivariate analysis. Connections between the orthogonal group 0" and the multivariate normal distribution are explored in James (1954) and in Wijsman (1957), as well as in many texts on multivariate analysis. In this chapter, invariance arguments are used to derive the density of a subblock of a uniformly distributed element of 0". This result is used to describe an upper bound on the rate at which one has convergence (as n ~ co) to the multivariate normal distribution.

The purpose of this chapter is to introduce the ideas surrounding the so called finite de Finetti style theorems. Four examples, one of which comes from the classical de Finetti theorem and three related to the normal distribution, are discussed here. These examples are introduced by first describing the "infinite version" of a result and then moving to the "finite version." In all of these examples, the infinite version came first, followed by a finite version. However, recent work on finite versions has suggested new infinite versions; some of these are discussed in the next chapter.

The implications of invariance assumptions for linear models are investigated here via a finite de Finetti style theorem. Before discussing linear models, we describe a general method for approximating projected measures. Of course, the origins of the method are in the four examples described in Chapter 8. All of the material in this chapter is from Diaconis, Eaton and Lauritzen (1987).