The Annals of Statistics

Quadratic distances on probabilities: A unified foundation

Bruce G. Lindsay, Marianthi Markatou, Surajit Ray, Ke Yang, and Shu-Chuan Chen

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This work builds a unified framework for the study of quadratic form distance measures as they are used in assessing the goodness of fit of models. Many important procedures have this structure, but the theory for these methods is dispersed and incomplete. Central to the statistical analysis of these distances is the spectral decomposition of the kernel that generates the distance. We show how this determines the limiting distribution of natural goodness-of-fit tests. Additionally, we develop a new notion, the spectral degrees of freedom of the test, based on this decomposition. The degrees of freedom are easy to compute and estimate, and can be used as a guide in the construction of useful procedures in this class.

Article information

Ann. Statist. Volume 36, Number 2 (2008), 983-1006.

First available in Project Euclid: 13 March 2008

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Zentralblatt MATH identifier

Primary: 62A01: Foundations and philosophical topics 62E20: Asymptotic distribution theory
Secondary: 62H10: Distribution of statistics

Degrees of freedom diffusion kernel goodness of fit high dimensions model assessment quadratic distance spectral decomposition


Lindsay, Bruce G.; Markatou, Marianthi; Ray, Surajit; Yang, Ke; Chen, Shu-Chuan. Quadratic distances on probabilities: A unified foundation. Ann. Statist. 36 (2008), no. 2, 983--1006. doi:10.1214/009053607000000956.

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