Approximating $f$-Divergences with Rank Statistics

J. M. de Frutos Porras

We introduce a rank-statistic approximation of $f$-divergences that avoids explicit density-ratio estimation by working directly with rank distributions. For a resolution parameter $K$, the mismatch between two univariate distributions $\mu$ and $\nu$ is mapped to a rank histogram on $\{0,\ldots,K\}$, whose deviation from uniformity is measured through a discrete $f$-divergence. We show that this estimator is monotone in $K$, provides a lower bound on the true $f$-divergence, and converges quantitatively as $K\to\infty$ under mild regularity assumptions. We extend the construction to high dimensions through random projections, establish convergence, deviation, and asymptotic normality results, and validate the method empirically in divergence estimation and generative modelling.

Palabras clave: f-divergences, rank statistics, divergence estimation, density-ratio-free methods,

Estadística No Paramétrica
3 de septiembre de 2026  11:10
Aula 29

A. Dorado Valín, A. López-Cheda, M. A. Jácome Pumar

L. Fernández Piana, V. Moreno, M. Svarc, A. Justel Eusebio

A. García Nogales