Decomposition and Invariance of Measures, and Statistical Transformation Models

Frontmatter

1. Introduction

Abstract

Decomposition or disintegration of measures and construction of invariant measures play essential roles in mathematics and in various fields of applied mathematics. In particular, the mathematical methodology in question is required for certain advanced parts of parametric statistics. See, for instance, Fraser (1979), Muirhead (1982), Barndorff-Nielsen, Blæsild, Jensen and Jørgensen (1982), Eaton (1983), Baddeley (1983), Andersson, Brøns and Jensen (1983), Farrell (1985), and Barndorff-Nielsen (1983, 1988).

Ole E. Barndorff-Nielsen, Preben Blæsild, Poul Svante Eriksen

2. Topological groups and actions

Abstract

In this section we introduce the concept of an action of a topological group G on a topological space χ. Furthermore we lay down a set of topological conditions to ensure that there exists a measure on χ which is invariant under the action of G.

Ole E. Barndorff-Nielsen, Preben Blæsild, Poul Svante Eriksen

3. Matrix Lie groups

Abstract

This section contains a brief introduction to matrix Lie groups. The focus is on developing tools for the construction of factorizations of a group with respect to some subgroup. In this connection the Lie algebra and the exponential map are the central concepts. It may be noted that all the groups occurring in the examples considered in these notes are matrix Lie groups.

Ole E. Barndorff-Nielsen, Preben Blæsild, Poul Svante Eriksen

4. Invariant, relatively invariant, and quasi-invariant measures

Abstract

In this section we discuss existence and uniqueness of invariant, relatively invariant and quasi-invariant measures on a space χ with an acting group G. In particular, the left and right invariant measures on G itself are considered, and several basic formulas relating these are derived. Various disintegration formulas are also presented.

Ole E. Barndorff-Nielsen, Preben Blæsild, Poul Svante Eriksen

5. Decomposition and factorization of measures

Abstract

Suppose a space χ is partitioned into disjoint subsets χ_π,_π∈ П, and let μ be a measure on χ. If for each _π∈ П we have a measure ρ_π on χ_π and if there is a measure κ on П such that \(\mathop \smallint \limits_X f\left( x \right)d\mu \left( x \right) = \mathop \smallint \limits_\Pi \mathop \smallint \limits_{{x_\pi }} f\left( x \right)d{\rho _\pi }\left( x \right)d\kappa \left( \pi \right)\) for every integrable function f then ((ρ_π)_π∈П,κ) is said to constitute a decomposition of μ, and we speak of (5.1) as a disintegration formula.

Ole E. Barndorff-Nielsen, Preben Blæsild, Poul Svante Eriksen

6. Construction of invariant measures

Abstract

We shall discuss here methods of constructing a G-invariant measure µ on the space χ, in the sense of expressing µ on the form \(\chi \left( {k,x} \right) = 1,x \in \chi ,k \in {G_x}.\)

Ole E. Barndorff-Nielsen, Preben Blæsild, Poul Svante Eriksen

7. Exterior calculus

Abstract

The exterior calculus of differential geometry provides procedures for factorization of measures and for the construction of invariant measures, which in many cases constitute a shortcut to the result. We wish here to indicate the technique so as to enable the reader to apply it without having to study exterior calculus as such. Accordingly, the discussion will in the present section be somewhat informal in comparison with the previous sections. For a comprehensive and rigorous exposition of exterior calculus see, for instance, Edelen (1985).

Ole E. Barndorff-Nielsen, Preben Blæsild, Poul Svante Eriksen

8. Statistical transformation models

Abstract

Many of the most important models in statistics are transformation models, or partly transformational. In this chapter we present the key aspects of the powerful theory of transformation models. This theory draws heavily on the theory of decomposition and invariance of measures considered in sections 2–7.

Ole E. Barndorff-Nielsen, Preben Blæsild, Poul Svante Eriksen

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