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-maxclosure 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.
Keywords
total positivity, kernel density estimation, MTP$_2$
