Somas aleatórias Poisson misturadas e modelos normais misturados associados

Abstract

The sum of a random number of random variables, besides being interesting from a probabilistic point of view, appears in applications involving processes that evolve over time. An important and widely studied example is the geometric sum that has many applications and models many phenomenas in insurance, finance, reliability, biology, among others. Motivated by the applicability and stochastic results of the geometric sum, in this work we obtain the limit distribution for partial sums with a random number of terms following a class of mixed Poisson distributions. The resulting weak limit is a mixing between a Normal distribution and an exponential family, which we call by Normal Exponential Family mixed (NEFM) laws. A new stability concept is introduced and a relationship with alpha-stable distributions is established. We propose estimation of the parameters of the NEFM models through method of moments and maximum likelihood method via EM-algorithm. In addition, we studied a limit distribution of the fractional Poisson sum, defining the normal Mittag-Leffler distribution, which is a mixture between the Normal and Mittag-Leffler distributions. We discussed the estimation of the parameters for this model through the Method of Moments and we found the asymptotic distribution of the estimators. Monte Carlo simulation studies are addressed to check the performance of the proposed estimators and two empirical illustrations on financial market and for precipitation data are presented.