Univariate and Bivariate Compound Geometric Gaussian Distributions

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Compound geometric random variables have been widely applied in the financial and actuarial disciplines. In these applications, the random variables subject to random sums are almost always non-negative. However, we propose a family of compound geometric distributions that deviates from this common paradigm. Consider a compound geometric Gaussian distribution, that is, a geometric sum of zero-mean Gaussian random variables. With an added location parameter and a particular normalization factor based on the geometric parameter, this forms a continuum of distributions between the Laplace and Gaussian families with convenient properties and only three parameters. These properties include easy interpretation of the parameters. In this work, we explore the characteristics, density functions, parameter interpretations and estimation techniques, and possible applications for this family. In addition, we investigate two 5-parameter bivariate versions of the family that combine the flexibilities of both the corresponding bivariate Laplace and Gaussian families. We exhibit some of the properties, examples, and parameter interpretation and estimation techniques for both families.

Compound geometric

Gaussian

multivariate

random sums of random variables 

IMS