Abstract
In this paper, we consider a discrete-time risk model with the claim number following a Poisson ARCH process. In this model, the mean of the current claim number depends on the previous observations. We study the large deviations for the aggregate amount for claims. For a heavy-tailed case, we obtain a precise large deviation formula, which agrees with existing ones in the literature. In computing the moderate deviation principle required by the structure of the claim-number process, our treatment substantially relies on an algorithm specifically designed for the autoregressive structure of our models.
Highlights
The goal of this paper is studying the precise large deviations for the aggregate claims n Nt Sn = Xt,j, t= j= ( . )where Nt is the number of claims in period t and {Xt,j, j =, . . . , t =, . . . , n} form an array of independent identically distributed (i.i.d.) claim-size random variables independent of Nt with distribution FX = – FX
Where Nt is the number of claims in period t and {Xt,j, j =, . . . , t =, . . . , n} form an array of independent identically distributed (i.i.d.) claim-size random variables independent of Nt with distribution FX = – FX
In the last few years, research on the time series models for count data has become a popular topic in the literature
Summary
Cossette et al [ ] used two integer-valued time series, Yu Journal of Inequalities and Applications (2016) 2016:140 namely the Poisson moving average (MA) and Poisson autoregressive (AR) processes, to model the claim frequency in the risk model. Li [ ] proposed a discrete-time risk model with the claim number being an integer-valued ARCH (INARCH) process with Poisson deviates, namely the model There is a vast amount of literature studying the asymptotic behavior of the large deviation of the risk models in the presence of heavy-tailed claim sizes. The precise large deviations to the risk model with the claim number being a Poisson ARCH process has not been considered in the literature. We establish the moderate deviation principle (MDP) for the partial sum n t=
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