Abstract
In this paper, we consider the Wiener–Poisson risk model, which consists of a Wiener process and a compound Poisson process. Given the discrete record of observations, we use a threshold method and a regularized Laplace inversion technique to estimate the survival probability. In addition, we also construct an estimator for the distribution function of jump size and study its consistency and asymptotic normality. Finally, we give some simulations to verify our results.
Highlights
Using {|∆i X |; 0 ≤ i ≤ n, IC n (θ(hn )) = 1} and empirical distribution function, i we can try to construct an estimator of F as follows: Lhbn, Fn (u) =
We use the threshold estimation technique and regularized Laplace inversion technique to constructed an estimator of survival probability for the Wiener–Poisson risk model
We will combine the threshold estimation technique with Fourier transform technique to construct an estimator of survival probability
Summary
∑ γi , t ≥ 0, i =1 where { Nt }t≥0 is a Poisson process with unknown intensity λ > 0, and γ1 , γ2 , γ3 , ... are independent and identically distributed positive sequence of random variables with unknown distribution function. In [17], the author assumed that { Xtn |tin = ihn ; i = 0, 1, 2, ..., n} and {γ1 , γ2 , ..., γ Ntn } are i n observed, where hn = tin − tin−1 is the sampling interval and the time of claims are known. The author constructed an estimator for Gerber–Shiu function and obtained its asymptotic property. Given the discrete record of observations, we need to judge whether a claim occurs in the interval In [14,17], the authors estimated the ruin probability and Gerber–Shiu function by a regularized. Using the threshold method and the work in [23], it is easy to obtain an estimator for the Laplace transform of Φ( x ).
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