Journal
ANNALS OF STATISTICS
Volume 36, Issue 2, Pages 983-1006Publisher
INST MATHEMATICAL STATISTICS
DOI: 10.1214/009053607000000956
Keywords
degrees of freedom; diffusion kernel; goodness of fit; high dimensions; model assessment; quadratic distance; spectral decomposition
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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.
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