Quadratic distances on probabilities: A unified foundation

TL;DR

This paper develops a unified theoretical framework for quadratic form distance measures used in goodness-of-fit testing, introducing spectral decomposition and spectral degrees of freedom to improve understanding and application.

Contribution

It provides a comprehensive, spectral-based foundation for quadratic distances, unifying dispersed theories and introducing spectral degrees of freedom for better test construction.

Findings

Spectral decomposition determines the limiting distribution of goodness-of-fit tests.

Introduction of spectral degrees of freedom as a practical tool.

Framework applicable to various quadratic distance procedures.

Abstract

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.