Nonparametric Estimation of Renyi Divergence and Friends
Abstract
We consider nonparametric estimation of $L_2$, Renyi$\alpha$ and Tsallis$\alpha$ divergences between continuous distributions. Our approach is to construct estimators for particular integral functionals of two densities and translate them into divergence estimators. For the integral functionals, our estimators are based on corrections of a preliminary plugin estimator. We show that these estimators achieve the parametric convergence rate of $n^{1/2}$ when the densities' smoothness, $s$, are both at least $d/4$ where $d$ is the dimension. We also derive minimax lower bounds for this problem which confirm that $s > d/4$ is necessary to achieve the $n^{1/2}$ rate of convergence. We validate our theoretical guarantees with a number of simulations.
 Publication:

arXiv eprints
 Pub Date:
 February 2014
 arXiv:
 arXiv:1402.2966
 Bibcode:
 2014arXiv1402.2966K
 Keywords:

 Statistics  Machine Learning;
 Mathematics  Statistics Theory