We study the limit distribution of partial sums with a random number of terms following a class of mixed Poisson distributions. The resulting weak limit is a mixture between a normal distribution and an exponential family, which we call by normal exponential family (NEF) laws. A new stability concept is introduced and a relationship between α-stable distributions and NEF laws is established. We propose the estimation of the NEF model parameters through the method of moments and also by the maximum likelihood method via an Expectation–Maximization algorithm. Monte Carlo simulation studies are addressed to check the performance of the proposed estimators, and an empirical illustration of the financial market is presented.
Bibliographical noteKAUST Repository Item: Exported on 2020-11-28
Acknowledgements: We would like to express our gratitude to the referee for the insightful comments, suggestions, and careful reading which lead to an improvement in the presentation of our paper. G. Oliveira thanks the financial support from Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES-Brazil). W. Barreto-Souza acknowledges support for his research from the KAUST Research Fund, NIH 1R01EB028753-01, and the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq-Brazil, Grant Number 305543/2018-0). This work is part of the Ph.D. thesis of Gabriela Oliveira realized at the Department of Statistics of the Universidade Federal de Minas Gerais, Brazil.