## Quadratic distances on probabilities: A unified foundation.(English)Zbl 1133.62001

Summary: This work builds a unified framework for the study of quadratic form distance measures as they are used in assessing 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.

### MSC:

 62A01 Foundations and philosophical topics in statistics 62E20 Asymptotic distribution theory in statistics 62H05 Characterization and structure theory for multivariate probability distributions; copulas 62G10 Nonparametric hypothesis testing

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