Kernel Density Estimation for Totally Positive Random Vectors
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We study the estimation of the density of a totally positive random vector. Total positivity of the distribution of a vector implies a strong form of positive dependence between its coordinates, and in particular, it implies positive association. We take on a modified kernel density estimation approach for estimating such a totally positive density. 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 provide experimental results to demonstrate the improved convergence of our modified kernel density estimator over the regular kernel density estimator, conjecturing that augmenting our sample with all points from its min-max closure relieves the curse of dimensionality.
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