Decomposition and invariance of measures, and statistical transformation models.

*(English)*Zbl 1088.62500
Lecture Notes in Statistics 58. New York etc.: Springer-Verlag (ISBN 0-387-97131-9). v, 147 p. (1989).

From the introduction: 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. However, a comprehensive exposition of the relevant mathematical results is not available in the statistical literature, nor in the mathematical, and where the various results can be found they are often rather inaccessible, being included in some advanced and more comprehensive mathematical treatises.

The present motes constitute an attempt to remedy this situation somewhat, particularly as concerns the need in statistics. An important starting base for the work has been an excellent set of notes by S. Andersson [Invariant measures. Tech. Rep. No. 129, Dept. Statistics, Standord Univ. (1978)]. We have tried to strike a suitable balance between a complete account of the mathematical theory and a mere skeleton of formulas, thus providing the interested reader with enough details and references to enable him to complete the account mathematically, if he so desires. The reader is assumed to have some, rather limited, elemental knowledge of topology, group theory and differential geometry, and – if he or she wishes to study the statistical applications in section 8 – a considerable knowledge of parametric statistics.

In section 2 we discuss the actions of groups on spaces, with the ensuing concepts of orbital decompositions and right and left factorizations of groups. Some basic results on matrix Lie groups and the associated Lie algebras are provided in section 3. Section 4 contains the definitions of invariant, relatively invariant and quasi-invariant measures and conditions for the existence of such measures, while methods for constructing invariant measures are considered in section 6. Part of the material of section 6 is intimately connected to questions of decomposition and factorization of measures which is the subject of section 5. The exterior calculus of differential geometry often provides an efficient and elegant way of decomposing a measure or finding an invariant measure. The most relevant aspects of exterior calculus are outlined in section 7. The final section 8 illustrates the usefulness of the mathematical tools by deriving the key properties of statistical transformation models. That section has been organized with the aim of enabling a reader with only a limited knowledge of the material in sections 2–7 to follow the main lines of the developments. However, a good general knowledge of parametric statistics is needed for a full appreciation of the discussion in section 8.

Another area of statistics in which decomposition, or disintegration, of measures is of great importance is that of spatial statistics. However, except for a derivation of the Blaschke-Petkantschin formula, we do not touch upon that area here. In particular, we do not discuss Palm measures and Gibbs kernels.

Examples are considered throughout the book, and a small collection of further results and exercises is included at the end. In particular, an outline of the main properties of exponential transformation models is given as an exercise. (Most of the examples in section 8 are, in fact, concerned with models of this type.)

Some of the results discussed are novel, but most of the material stems from the existing literature. The relation of the material presented here to other parts of the literature is indicated in the bibliographical notes which conclude each of the sections 2–8.

Contents: 1. Introduction; 2. Topological groups and actions; 3. Matrix Lie groups; 4. Invariant, relatively invariant, and quasi-invariant measures; 5. Decomposition and factorization of measures; 6. Construction of invariant measures; 7. Exterior calculus; 8. Statistical transformation models.

Further results and exercises; References, with author index; Subject index; Notation index.

The present motes constitute an attempt to remedy this situation somewhat, particularly as concerns the need in statistics. An important starting base for the work has been an excellent set of notes by S. Andersson [Invariant measures. Tech. Rep. No. 129, Dept. Statistics, Standord Univ. (1978)]. We have tried to strike a suitable balance between a complete account of the mathematical theory and a mere skeleton of formulas, thus providing the interested reader with enough details and references to enable him to complete the account mathematically, if he so desires. The reader is assumed to have some, rather limited, elemental knowledge of topology, group theory and differential geometry, and – if he or she wishes to study the statistical applications in section 8 – a considerable knowledge of parametric statistics.

In section 2 we discuss the actions of groups on spaces, with the ensuing concepts of orbital decompositions and right and left factorizations of groups. Some basic results on matrix Lie groups and the associated Lie algebras are provided in section 3. Section 4 contains the definitions of invariant, relatively invariant and quasi-invariant measures and conditions for the existence of such measures, while methods for constructing invariant measures are considered in section 6. Part of the material of section 6 is intimately connected to questions of decomposition and factorization of measures which is the subject of section 5. The exterior calculus of differential geometry often provides an efficient and elegant way of decomposing a measure or finding an invariant measure. The most relevant aspects of exterior calculus are outlined in section 7. The final section 8 illustrates the usefulness of the mathematical tools by deriving the key properties of statistical transformation models. That section has been organized with the aim of enabling a reader with only a limited knowledge of the material in sections 2–7 to follow the main lines of the developments. However, a good general knowledge of parametric statistics is needed for a full appreciation of the discussion in section 8.

Another area of statistics in which decomposition, or disintegration, of measures is of great importance is that of spatial statistics. However, except for a derivation of the Blaschke-Petkantschin formula, we do not touch upon that area here. In particular, we do not discuss Palm measures and Gibbs kernels.

Examples are considered throughout the book, and a small collection of further results and exercises is included at the end. In particular, an outline of the main properties of exponential transformation models is given as an exercise. (Most of the examples in section 8 are, in fact, concerned with models of this type.)

Some of the results discussed are novel, but most of the material stems from the existing literature. The relation of the material presented here to other parts of the literature is indicated in the bibliographical notes which conclude each of the sections 2–8.

Contents: 1. Introduction; 2. Topological groups and actions; 3. Matrix Lie groups; 4. Invariant, relatively invariant, and quasi-invariant measures; 5. Decomposition and factorization of measures; 6. Construction of invariant measures; 7. Exterior calculus; 8. Statistical transformation models.

Further results and exercises; References, with author index; Subject index; Notation index.