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Convolutions of totally positive distributions with applications to kernel density estimation

Ali Zartash and Elina Robeva

Vol. 13 (2022), No. 1, 57–79

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.

total positivity, kernel density estimation, MTP$_2$
Mathematical Subject Classification
Primary: 62G05, 62G07, 62R01
Secondary: 62H20
Received: 26 December 2020
Revised: 17 January 2022
Accepted: 24 January 2022
Published: 4 May 2023
Ali Zartash
Massachusetts Institute of Technology
Cambridge, MA
United States
Elina Robeva
University of British Columbia
Vancouver, BC