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Estimation of multivariate generalized gamma convolutions through Laguerre expansions

Oskar Laverny, Esterina Masiello, Véronique Maume-Deschamps, Didier Rullière

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 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.

Paper Structure

This paper contains 24 sections, 2 theorems, 52 equations, 13 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Gamma convolutions classes
  3. Definitions and first properties
  4. Shifted moments and shifted cumulants
  5. Projection from G1 to G1n
  6. Construction of an estimator in Gdn
  7. The tensorized Laguerre basis
  8. Exponential decay of Laguerre coefficients
  9. Consistency of the induced empirical loss
  10. Investigation of performance
  11. Projection from a known density
  12. Projection from G1 densities to G1n
  13. Projection from densities outside G1 to G1n
  14. Estimation from univariate empirical data
  15. Heavy-tailed examples
  16. ...and 9 more sections

Key Result

Theorem 3.2.1

For any $\varepsilon, \varepsilon'$ s.t. $\varepsilon > \varepsilon' > 0$ and any dimension $d$, there exists a finite positive constant $B(d,\varepsilon')$ such that the Laguerre coefficients $\left(a_{\bm k}\right)_{\bm k \in \mathbb N^d}$ of any $d$-variate $\varepsilon$-well-behaved gamma convol

Figures (13)

  • Figure 1: Violin densities of resampled KS distances (smaller is better). In abscissa is $n \in \{2,3,4,5,10,20,30,40\}$, the number of gammas used in the approximation.
  • Figure 2: Violin densities of resampled KS distances for approximation of a Weibull($k=1.5$). In abscissa is $n \in \{2,3,4,5,10,20,30,40\}$, the number of gammas used in the approximation. Only the Laguerre approximation results are presented.
  • Figure 3: Log-Normal results with $10$ gammas: Upper left: the comparison of the density approximations. Lower left: the difference of to the true density. Upper right: a quantile-quantile plot of the approximation against the true distribution. Lower right: p-values of one-sample KS tests of simulations from the estimator against the true distribution.
  • Figure 4: Quantile-quantile plots for Pareto experiments, with $N=1000$ samples (log-scaled). We only show the $995$ first points: $5$ points are excluded in the tail for clarity. Each row corresponds to a shape parameter for the Pareto, and each column to a number of gammas.
  • Figure 5: Weibull results with $2,10,20$ and $40$ gammas. Same legend as Figure \ref{['fig:log-Normal10']}.
  • ...and 8 more figures

Theorems & Definitions (27)

  • Definition 2.1.1: $\mathcal{G}_{1,n}$ and $\mathcal{G}_{1}$
  • Definition 2.1.3: $\mathcal{G}_{d,n}$ and $\mathcal{G}_{d}$
  • proof
  • Example 2.1.5: A curious distribution
  • proof
  • Remark 2.1.6: The no-bijection result
  • Definition 2.2.1: Shifted moments and cumulants
  • proof
  • Definition 3.1.1: Laguerre function
  • Example 3.1.2: $\bm X \sim \mathcal{G}_{d,1}(\alpha,\bm s)$
  • ...and 17 more