Abstract
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.
Highlights
Olof Thorin introduced the Generalized Gamma Convolutions (GGC) class of distributions in 1977 as a tool to study the infinite divisibility of Pareto [56] and log-Normal [57] distributions
We show that the theoretical background of the estimation procedure in [41, 20] is inherently univariate, and no direct extension to a multivariate case is possible
The Laguerre coefficients a =k≤m overflow the Float64 limits, but the implementation we provide in the Julia package ThorinDistributions.jl [32], ensures that the computations do not overflow by using multiple precision arithmetic when needed
Summary
Olof Thorin introduced the Generalized Gamma Convolutions (GGC) class of distributions in 1977 as a tool to study the infinite divisibility of Pareto [56] and log-Normal [57] distributions. We introduce a new quantification of the “well-behavior” of a multivariate gamma convolution, that we show to be equivalent to the exponential decay of Laguerre coefficients Using this quantification, we bridge the gaps and provide a new stable algorithm for the evaluation of theoretical densities in both the univariate and multivariate cases, as well as consistent parameter estimation procedures which handle both clean data (the density to be estimated is given as a formal function) and dirty data (such as an empirical dataset in 64 bits precision).
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
