Divergence Measures Estimation and Its Asymptotic Normality Theory Using Wavelets Empirical Processes

TL;DR

This paper develops the asymptotic theory for divergence measures like Renyi, Tsallis, and Kullback-Leibler using wavelet-based empirical processes, providing new statistical tests and convergence results.

Contribution

It introduces a wavelet-based approach to estimate divergence measures and derives their asymptotic properties, including normality and convergence rates, for the first time.

Findings

Derived asymptotic normality for divergence estimators

Established convergence rates in Besov spaces

Developed statistical tests based on divergence measures

Abstract

In this paper we provide the asymptotic theory of the general of $\phi$-divergences measures, which includes the most common divergence measures : Renyi and Tsallis families and the Kullback-Leibler measure. Instead of using the Parzen nonparametric estimators of the probability density functions whose discrepancy is estimated, we use the wavelets approach and the geometry of Besov spaces. One-sided and two-sided statistical tests are derived as well as symmetrized estimators. Almost sure rates of convergence and asymptotic normality theorem are obtained in the general case, and next particularized for the Renyi and Tsallis families and for the Kullback-Leibler measure as well. The applicability of the results to usual distribution functions is addressed.