Estimation of multivariate generalized gamma convolutions through Laguerre expansions.

Abstract

The generalized gamma convolutions class of distributions appeared in Thorin's work while looking for the infinite divisibility of the log-Normal and Pareto distributions. Although these distributions have been extensively studied in the univariate case, the multivariate case and the dependence structures that can arise from it have received little interest in the literature. Furthermore, only one projection procedure for the univariate case was recently constructed, and no estimation procedures are available. By expanding the densities of multivariate generalized gamma convolutions into a tensorized Laguerre basis, we bridge the gap and provide performant estimation procedures for both the univariate and multivariate cases. We provide some insights about performance of these procedures, and a convergent series for the density of multivariate gamma convolutions, which is shown to be more stable than Moschopoulos's and Mathai's univariate series. We furthermore discuss some examples.

Estimation of high dimensional Gamma convolutions through random projections

“…Miles, Furman and Kuznetsov [25,36] provide a way to project a known density onto the univariate class, but the procedure is somewhat complicated and requires very precise integration of shifted moments. Up until [31] recently, there were no estimation procedure for distributions in this class, neither univariate nor multivariate. This estimation procedure uses a projection in a Laguerre basis [23,32,15], together with a bijection between parameters of the distribution and its coefficients in the basis, to produce an efficient loss and a fast numerical procedure to fit multivariate generalized Gamma convolutions.…”

“…• Section 2 and Section 3 give definitions of the generalized Gamma convolutions, fix some notations, describe the approach from [31] which we leverage, and discuss briefly Grassmannian cubatures and random projections of moments problems.…”

“…Mataï [33] and later Moschopoulos [38] provide convergent series to estimate the densities of G 1 distributions with finite Thorin measure. In [31], a more systematic approach is given to the multivariate density problem through Laguerre expansions, generalizing Moschopoulos's series to a multivariate case. DEFINITION 2.3 (Laguerre basis [31,23,15,32]).…”

“…In [31], a more systematic approach is given to the multivariate density problem through Laguerre expansions, generalizing Moschopoulos's series to a multivariate case. DEFINITION 2.3 (Laguerre basis [31,23,15,32]). The Laguerre functions…”

“…Following [31], we can therefore approximate the integrated square error between the densities through its projection in the truncated Laguerre basis, which provides the following empirical loss for a candidate Thorin measure ν:…”