Convergence and inference for mixed Poisson random sums
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Convergence and inference for mixed Poisson random sums Gabriela Oliveira1 · Wagner Barreto-Souza1,2
· Roger W. C. Silva1
Received: 3 June 2020 / Accepted: 4 November 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract 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 αstable 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. Keywords EM-algorithm · Mixed Poisson distribution · Stability · Weak convergence Mathematics Subject Classification 60F05 · 62E20 · 62Fxx
1 Introduction One of the most important and beautiful theorems in probability theory is the Central Limit Theorem, which lays down the convergence in distribution of the partial sum (properly normalized) of i.i.d. random variables with finite second moment to a normal distribution. This can be seen as a characterization of the normal distribution as the weak limit of such sums. A natural variant of this problem is placed when the number of terms in the sum is random. For instance, counting processes are of fundamental
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Wagner Barreto-Souza [email protected] Gabriela Oliveira [email protected] Roger W. C. Silva [email protected]
1
Departamento de Estatística, Universidade Federal de Minas Gerais, Belo Horizonte, Brazil
2
Statistics Program, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia
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G. Oliveira et al.
importance in the theory of probability and statistics. A comprehensive account for this topic is given in Gnedenko and Korolev (1996). One of the earliest counting models is the compound Poisson process {Ct }t≥0 defined as Ct =
Nt
X n , t ≥ 0,
(1)
n=1
where {Nt }t≥0 is a Poisson process with rate λt, λ > 0, and {X n }n∈N is a sequence of i.i.d. random variables independent of {Nt }t≥0 . Applications of the random summation (1) include risk theory, biology, queuing theory and finance; for instance, see Embrechts et al. (2003), Paulsen (2008) and Puig and Barquinero (2010). For fixed t, it can be shown that the random summation given in (1), when properly normalized, converges weakly to the standard normal distribution as λ → ∞. Another important quantity is the geometric random summation defined as Sp =
νp
Xn,
n=1
where ν p is a geometric random variable with probability function P(ν p = k) = (1 − p)k−1 p, k = 1, 2, . . . , and {X n }n∈N is a sequence of i.i.d. random variables independent of ν p , for p ∈ (0, 1). Geometric summation has
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