The limiting behavior of some infinitely divisible exponential dispersion models

Consider an exponential dispersion model (EDM) generated by a probability $\mu$ on $[0,\infty )$ which is infinitely divisible with an unbounded L\'{e}vy measure $\nu$. The Jorgensen set (i.e., the dispersion parameter space) is then $\mathbb{R}^{+}$, in which case the EDM is characterized by two parameters: $\theta _{0}$ the natural parameter of the associated natural exponential family and the Jorgensen (or dispersion) parameter $t$. Denote by $EDM(\theta _{0},t)$ the corresponding distribution and let $Y_{t}$ is a r.v. with distribution $EDM(\theta_0,t)$. Then if $\nu ((x,\infty ))\sim -\ell \log x$ around zero we prove that the limiting law $F_0$ of $Y_{t}^{-t}$ as $t\rightarrow 0$ is of a Pareto type (not depending on $\theta_0$) with the form $F_0(u)=0$ for $u<1$ and $1-u^{-\ell }$ for $u\geq 1$. Such a result enables an approximation of the distribution of $Y_{t}$ for relatively small values of the dispersion parameter of the corresponding EDM. Illustrative examples are provided.Comment: 8 page

Monte Carlo Methods for Insurance Risk Computation

In this paper we consider the problem of computing tail probabilities of the distribution of a random sum of positive random variables. We assume that the individual variables follow a reproducible natural exponential family (NEF) distribution, and that the random number has a NEF counting distribution with a cubic variance function. This specific modelling is supported by data of the aggregated claim distribution of an insurance company. Large tail probabilities are important as they reflect the risk of large losses, however, analytic or numerical expressions are not available. We propose several simulation algorithms which are based on an asymptotic analysis of the distribution of the counting variable and on the reproducibility property of the claim distribution. The aggregated sum is simulated efficiently by importancesampling using an exponential cahnge of measure. We conclude by numerical experiments of these algorithms.Comment: 26 pages, 4 figure

On the small-time behavior of subordinators

We prove several results on the behavior near t=0 of $Y_t^{-t}$ for certain $(0,\infty)$-valued stochastic processes $(Y_t)_{t>0}$. In particular, we show for L\'{e}vy subordinators that the Pareto law on $[1,\infty)$ is the only possible weak limit and provide necessary and sufficient conditions for the convergence. More generally, we also consider the weak convergence of $tL(Y_t)$ as $t\to0$ for a decreasing function $L$ that is slowly varying at zero. Various examples demonstrating the applicability of the results are presented.Comment: Published in at http://dx.doi.org/10.3150/11-BEJ363 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

New exponential dispersion models for count data -- the ABM and LM classes

In their fundamental paper on cubic variance functions, Letac and Mora (The Annals of Statistics,1990) presented a systematic, rigorous and comprehensive study of natural exponential families on the real line, their characterization through their variance functions and mean value parameterization. They presented a section that for some reason has been left unnoticed. This section deals with the construction of variance functions associated with natural exponential families of counting distributions on the set of nonnegative integers and allows to find the corresponding generating measures. As exponential dispersion models are based on natural exponential families, we introduce in this paper two new classes of exponential dispersion models based on their results. For these classes, which are associated with simple variance functions, we derive their mean value parameterization and their associated generating measures. We also prove that they have some desirable properties. Both classes are shown to be overdispersed and zero-inflated in ascending order, making them as competitive statistical models for those in use in both, statistical and actuarial modeling. To our best knowledge, the classes of counting distributions we present in this paper, have not been introduced or discussed before in the literature. To show that our classes can serve as competitive statistical models for those in use (e.g., Poisson, Negative binomial), we include a numerical example of real data. In this example, we compare the performance of our classes with relevant competitive models.Comment: 27 pages, 4 tables, 3 figure

Cumulant-Based Goodness-of-Fit Tests for the Tweedie, Bar-Lev and Enis Class of Distributions

The class of natural exponential families (NEFs) of distributions having power variance functions (NEF-PVFs) is huge (uncountable), with enormous applications in various fields. Based on a characterization property that holds for the cumulants of the members of this class, we developed a novel goodness-of-fit (gof) test for testing whether a given random sample fits fixed members of this class. We derived the asymptotic null distribution of the test statistic and developed an appropriate bootstrap scheme. As the content of the paper is mainly theoretical, we exemplify its applicability to only a few elements of the NEF-PVF class, specifically, the gamma and modified Bessel-type NEFs. A Monte Carlo study was executed for examining the performance of both—the asymptotic test and the bootstrap counterpart—in controlling the type I error rate and evaluating their power performance in the special case of gamma, while real data examples demonstrate the applicability of the gof test to the modified Bessel distribution