  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.
