Quadratic distances on probabilities: A unified foundation

Details Quadratic distances on probabilities: A unified foundation Quadratic distances on probabilities: A unified foundation

2008-04-07

arxiv.org/abs/0804.0991v1

Annals of Statistics 2008, Vol. 36, No. 2, 983-1006

Mathematics

Statistics Theory

Published in at http://dx.doi.org/10.1214/009053607000000956 the Annals of Statistics (http://www.imstat.org/aos/) by the Inst

Scientific paper

10.1214/009053607000000956

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.

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