Estimation of high dimensional Gamma convolutions through random projections

TL;DR

This paper introduces a stochastic estimation method for high-dimensional multivariate Gamma convolutions using random projections and advanced mathematical tools, enabling effective distribution fitting in high dimensions.

Contribution

It develops a novel stochastic estimation procedure leveraging random projections and mathematical analysis for fitting high-dimensional Gamma convolution distributions.

Findings

Convergence of stochastic gradient descent to accurate estimators

Effective distribution fitting in high-dimensional settings

Validation through examples in low and high dimensions

Abstract

Multivariate generalized Gamma convolutions are distributions defined by a convolutional semi-parametric structure. Their flexible dependence structures, the marginal possibilities and their useful convolutional expression make them appealing to the practitioner. However, fitting such distributions when the dimension gets high is a challenge. We propose stochastic estimation procedures based on the approximation of a Laguerre integrated square error via (shifted) cumulants approximation, evaluated on random projections of the dataset. Through the analysis of our loss via tools from Grassmannian cubatures, sparse optimization on measures and Wasserstein gradient flows, we show the convergence of the stochastic gradient descent to a proper estimator of the high dimensional distribution. We propose several examples on both low and high-dimensional settings.