Abstract

We study the problem of estimating 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. We take a (modified) kernel density estimation approach to estimate a totally positive density. Our main result is that the sum of scaled standard Gaussians 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 Gaussians 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.

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