The Theory of Canonical Moments with Applications in Statistics, Probability, and Analysis

This new material is concerned with the theory and applications of probability, statistics and analysis of canonical moments. It provides a powerful tool for the determination of optimal experimental designs, for the calculation of the main characteristics of random walks, and for other moment problems appearing in probability and statistics.Abweichend von dem in der Literatur üblichen Ansatz, wird die Momententheorie und ihre Anwendung hier aus dem Blickwinkel von Statistik, Wahrscheinlichkeitstheorie und Analysis betrachtet. Zweck des Buches ist aufzuzeigen, daß die kanonischen Momente ein sehr leistungsstarkes Instrument sind zur Bestimmung der optimalen Versuchsplanung, zur Berechnung der Hauptmerkmale der Random-Walk-Theorie und zur Behandlung wahrscheinlichkeits- und statistikspezifischer Momentproblematik. Die Themenauswahl erfolgte unter dem Gesichtspunkt, daß einerseits anwendungsorientierte Leser einen ausreichend großen Einblick gewinnen, um mit dieser Problematik ganz konkret arbeiten zu können und andererseits Theoretiker eine erschöpfende Darstellung des mathematischen Hintergrundes erhalten. (10/97) Canonical Moments.

Orthogonal Polynomials.

Continued Fractions and the Stieltjes Transform.

Special Sequences of Canonical Moments.

Canonical Moments and Optimal Design--First Applications.

Discrimination and Model Robust Designs.

Applications in Approximation Theory.

Canonical Moments and Random Walks.

The Circle and Trigonometric Functions.

Further Applications.

Bibliography.

Indexes. HOLGER DETTE is Professor of Mathematics at Ruhr-Universität Bochum, Fakultät und Institut für Mathematik, Germany.
WILLIAM J. STUDDEN is Professor of Statistics and Mathematics at Purdue University. The fascinating world of canonical moments--a unique look at this practical, powerful statistical and probability tool

Unusual in its emphasis, this landmark monograph on canonical moments describes the theory and application of canonical moments of probability measures on intervals of the real line and measures on the circle. Stemming from the discovery that canonical moments appear to be more intrinsically related to the measure than ordinary moments, the book's main focus is the broad application of canonical moments in many areas of statistics, probability, and analysis, including problems in the design of experiments, simple random walks or birth and death chains, and in approximation theory.

The book begins with an explanation of the development of the theory of canonical moments for measures on intervals [a, b] and then describes the various practical applications of canonical moments. The book's topical range includes:
* Definition of canonical moments both geometrically and as ratios of Hankel determinants
* Orthogonal polynomials viewed geometrically as hyperplanes to moment spaces
* Continued fractions and their link between ordinary moments and canonical moments
* The determination of optimal designs for polynomial regression
* The relationships between canonical moments, random walks, and orthogonal polynomials
* Canonical moments for the circle or trigonometric functions

Finally, this volume clearly illustrates the powerful mathematical role of canonical moments in a chapter arrangement that is as logical and interdependent as is the relationship of canonical moments to statistics, probability, and analysis.