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.
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.