Convolutions of Totally Positive Distributions with applications to Kernel Density Estimation

TL;DR

This paper investigates non-parametric density estimation for totally positive distributions, proposing a kernel-based method that guarantees total positivity through convolution with Gaussian kernels and exploring its theoretical properties.

Contribution

It introduces a kernel density estimator that preserves total positivity by convolving with Gaussian kernels and establishes a novel connection between min-max sets and total positivity.

Findings

Sum of Gaussian bumps over min-max closed sets yields totally positive distributions

Convolution with Gaussian preserves total positivity in certain cases

Proposes a practical method for estimating totally positive densities

Abstract

In this work we study the estimation of the density of a totally positive random vector. Total positivity of the distribution of a random vector implies a strong form of positive dependence between its coordinates and, in particular, it implies positive association. Since estimating a totally positive density is a non-parametric problem, we take on a (modified) kernel density estimation approach. Our main result is that the sum of scaled standard Gaussian bumps centered at a min-max closed set provably yields a totally positive distribution. Hence, our strategy for producing a totally positive estimator is to form the min-max closure of the set of samples, and output a sum of Gaussian bumps centered at the points in this set. We can frame this sum as a convolution between the uniform distribution on a min-max closed set and a scaled standard Gaussian. We further conjecture that convolving any totally positive density with a standard Gaussian remains totally positive.