Estimation of multivariate generalized gamma convolutions through Laguerre expansions

Oskar Laverny (1, 2, 3) , Esterina Masiello (2) , Véronique Maume-Deschamps (2) , Didier Rullière (4, 5, 6, 7)

1 PSPM - Probabilités, statistique, physique mathématique
2 ICJ - Institut Camille Jordan
3 SCOR SE
4 Mines Saint-Étienne MSE - École des Mines de Saint-Étienne
5 FAYOL-ENSMSE - Institut Henri Fayol
6 LIMOS - Laboratoire d'Informatique, de Modélisation et d'Optimisation des Systèmes
7 FAYOL-ENSMSE - Département Génie mathématique et industriel

The generalized gamma convolution class of distribution 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 procedure are available. By expending the densities of multivariate generalized gamma convolutions into a tensorized Laguerre basis, we bridge the gap and provide performant estimations 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.

Mots clés: Multivariate Generalized Gamma Convolutions; Estimation; Thorin's measure; Laguerre's basis; Infinite Divisibility

Domaines: Statistiques [math.ST]

Oskar Laverny, Esterina Masiello, Véronique Maume-Deschamps, Didier Rullière. Estimation of multivariate generalized gamma convolutions through Laguerre expansions. Electronic Journal of Statistics , 2021, 15 (2), pp.5158-5202. ⟨10.1214/21-EJS1918⟩. ⟨hal-03160289v2⟩