Asymptotically Normal Estimators of the Ruin Probability for Lévy Insurance Surplus from Discrete Samples

Abstract

A statistical inference for ruin probability from a certain discrete sample of the surplus is discussed under a spectrally negative Lévy insurance risk. We consider the Laguerre series expansion of ruin probability, and provide an estimator for any of its partial sums by computing the coefficients of the expansion. We show that the proposed estimator is asymptotically normal and consistent with the optimal rate of convergence and estimable asymptotic variance. This estimator enables not only a point estimation of ruin probability but also an approximated interval estimation and testing hypothesis.

Ultimate Time Survival Probability in Three-Risk Discrete Time Risk Model

“…In such a case, we can use also the upper estimate of ruin probability, which usually decreases with increasing initial capital. The useful estimates for the nonhomogeneous models we can find in [27][28][29][30][31] among others. For instance, results of [28,29] imply that ψ(u) c 1 exp{−c 2 u}, u 0, for all above examples with a positive constants c 1 , and c 2 depending on the numerical characteristics of the random claims X, Y and Z, generating a three-risk discrete time model.…”

The Gerber-Shiu discounted penalty function: A review from practical perspectives

“…Hence, every f ∈ L 2 ([0, +∞)) can be expanded as f (x) = ∑ k∈N 0 a f ,k ϕ k (x), with a f ,k := f , ϕ k = +∞ 0 f (x)ϕ k (x)dx. A possible advantage of employing the Laguerre series expansion is that the relevant estimator can be shown to be asymptotically normal, which is first discussed in [148] in the context of ruin probabilities under Lévy risk models. The Laguerre series expansion has found its application in obtaining the estimator of the Gerber-Shiu function under the Cramér-Lundberg model [193] with its asymptotic normality [154].…”

“…To this end, the first semi-parametric framework is developed in [143,144] via the regularized inversion of the Laplace transform (Section 5.3), whereas this approach is not satisfactory all the time due to the logarithmic convergence rate of the estimator [126]. Since then, a variety of statistical methods have been proposed based on the Fourier inversion [147] (Section 5.4), and series expansions using the Fourier-sinc/cosine series [165,166,183,191] and, more intensively, the Laguerre series [53,74,148,151,153,154,193,194] (Section 5.5), with which the rates of convergence are improved to polynomial orders. In addition, statistical methods based on series expansion are of particular interest as they have the potential to reveal the asymptotic law of the estimated Gerber-Shiu function.…”

Bi-seasonal discrete time risk model with income rate two

“…Scrolling across the timeline, an observable works of Gerber and Shiu on the risk collective models could be highlighted: [5], [6], [7] and [8]. Recently, many research papers on the related risk models as in (1) are occurring per year, see for example [9], [10], [11], [12], [13], [14], [15] and references therein.…”