Abstract
We propose a new stochastic model describing the joint distribution of (X,N), where N is a counting variable while X is the sum of N independent gamma random variables. We present the main properties of this general model, which include marginal and conditional distributions, integral transforms, moments and parameter estimation. We also discuss in more detail a special case where N has a heavy tailed discrete Pareto distribution. An example from finance illustrates the modeling potential of this new mixed bivariate distribution.
Highlights
We propose a new stochastic model describing the joint distribution of N (X, N) =d Xi, N, (1)i=1 where N is an integer-valued random variable on N = {1, 2, . . .} while the {Xi} are independent and identically distributed (IID) random variables on R+, independent of N
Our new model is a generalization of the Bivariate with exponential and geometric margins (BEG) and the Bivariate GammaGeometric (BGG) models introduced in Kozubowski and Panorska (2005) and Barreto-Souza (2012), respectively, where N was geometrically distributed with the probability mass function (PMF)
2 Bivariate episodes driven by an IID gamma sequence we summarize the properties which are common to all bivariate distributions describing the episodes (1) driven by a sequence {Xi} of IID gamma variables with the probability density function (PDF) (4), denoted by GAM(α, β)
Summary
In addition to presenting basic properties of the new model described above, we summarize fundamental properties of the entire class of such models with gamma distributed {Xi} and arbitrary distribution of the random variable N, which drives the dependence structure of the (X, N) vector in (1) These properties include infinite divisibility, the tail behavior of X, the conditional distributions of N given X = x as well as N given X > x, and estimation. Remark 1 In the special case where N is geometric (2) we recover the Bivariate GammaGeometric (BGG) distribution of Barreto-Souza (2012) In this case, the PDF of X can be written in terms of 2-parameter Mittag-Leffler (ML) special function.
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