On multivariate modifications of Cramer–Lundberg risk model with constant intensities

ABSTRACT

The paper considers very general multivariate modifications of Cramer–Lundberg risk model. The claims can be of different types and can arrive in groups. The groups arrival processes have constant intensities. The counting groups processes are dependent multivariate compound Poisson processes of Type I. We allow empty groups and show that in that case we can find stochastically equivalent Cramer–Lundberg model with non-empty groups. The investigated model generalizes the risk model with common shocks, the Poisson risk process of order k, the Poisson negative binomial, the Polya-Aeppli, the Polya-Aeppli of order k among others. All of them with one or more types of policies. The numerical characteristics, Cramer–Lundberg approximations, and probabilities of ruin are derived. During the paper, we show that the theory of these risk models intrinsically relates to the special types of integro differential equations. The probability solutions to such differential equations provide new insights, typically overseen from the standard point of view.     

A MCMC approach for modeling customer lifetime behavior using the COM‐Poisson distribution

One of the major challenges associated with the measurement of customer lifetime value is selecting an appropriate model for predicting customer future transactions. Among such models, the Pareto/negative binomial distribution (Pareto/NBD) is the most prevalent in noncontractual relationships characterized by latent customer defections; ie, defections are not observed by the firm when they happen. However, this model and its applications have some shortcomings. Firstly, a methodological shortcoming is that the Pareto/NBD, like all lifetime transaction models based on statistical distributions, assumes that the number of transactions by a customer follows a Poisson distribution. However, many applications have an empirical distribution that does not fit a Poisson model. Secondly, a computational concern is that the implementation of Pareto/NBD model presents some estimation challenges specifically related to the numerous evaluation of the Gaussian hypergeometric function. Finally, the model provides 4 parameters as output, which is insufficient to link the individual purchasing behavior to socio‐demographic information and to predict the behavior of new customers. In this paper, we model a customer's lifetime transactions using the Conway‐Maxwell‐Poisson distribution, which is a generalization of the Poisson distribution, offering more flexibility and a better fit to real‐world discrete data. To estimate parameters, we propose a Markov chain Monte Carlo algorithm, which is easy to implement. Use of this Bayesian paradigm provides individual customer estimates, which help link purchase behavior to socio‐demographic characteristics and an opportunity to target individual customers.     

Univariate and multivariate versions of the negative binomial-inverse Gaussian distributions with applications

In this paper we propose a new compound negative binomial distribution by mixing the p negative binomial parameter with an inverse Gaussian distribution and where we consider the reparameterization p=exp(−λ). This new formulation provides a tractable model with attractive properties which make it suitable for application not only in the insurance setting but also in other fields where overdispersion is observed. Basic properties of the new distribution are studied. A recurrence for the probabilities of the new distribution and an integral equation for the probability density function of the compound version, when the claim severities are absolutely continuous, are derived. A multivariate version of the new distribution is proposed. For this multivariate version, we provide marginal distributions, the means vector, the covariance matrix and a simple formula for computing multivariate probabilities. Estimation methods are discussed. Finally, examples of application for both univariate and bivariate cases are given.     

Approximation of inverse moments of discrete distributions

For a wide class of discrete distributions, we derive a representation of the inverse (negative) moments through the Stirling numbers of the first kind and inverse factorial moments. We specialize the results for the Poisson, binomial, hypergeometric and negative binomial distributions.     

On the first time of ruin in the bivariate compound Poisson model

This paper considers a bivariate compound Poisson model for a book of two dependent classes of insurance business. We focus on the ruin probability that at least one class of business will get ruined. As expected, general explicit expressions for this bivariate ruin probability is very difficult to obtain. In view of this, we introduce the so-called bivariate compound binomial model which can be used to approximate the finite-time survival probability of the assumed model. We then study some simple bounds for the infinite-time ruin probability via the association properties of the bivariate compound Poisson model. We also investigate the impact of dependence on the infinite-time ruin probability by means of multivariate stochastic orders.     

一类离散双险种风险模型

本研究了一类离散双险种风险模型，其中一类险种的索赔到达为泊松随机序列，另一类险种的索赔到达为二项随机序列．得到了最终破产概率的Lundberg不等式以及一般表达式．     

On the credibility of insurance claim frequency: Generalized count models and parametric estimators

We analyze the concept of credibility in claim frequency in two generalized count models–Mittag-Leffler and Weibull count models–which can handle both underdispersion and overdispersion in count data and nest the commonly used Poisson model as a special case. We find evidence, using data from a Danish insurance company, that the simple Poisson model can set the credibility weight to one even when there are only three years of individual experience data resulting from large heterogeneity among policyholders, and in doing so, it can thus break down the credibility model. The generalized count models, on the other hand, allow the weight to adjust according to the number of years of experience available. We propose parametric estimators for the structural parameters in the credibility formula using the mean and variance of the assumed distributions and a maximum likelihood estimation over a collective data. As an example, we show that the proposed parameters from Mittag-Leffler provide weights that are consistent with the idea of credibility. A simulation study is carried out investigating the stability of the maximum likelihood estimates from the Weibull count model. Finally, we extend the analyses to multidimensional lines and explain how our approach can be used in selecting profitable customers in cross-selling; customers can now be selected by estimating a function of their unknown risk profiles, which is the mean of the assumed distribution on their number of claims.     

A first–passage time random walk distribution with five transition probabilities: a generalization of the shifted inverse trinomial

In this paper a univariate discrete distribution, denoted by GIT, is proposed as a generalization of the shifted inverse trinomial distribution, and is formulated as a first-passage time distribution of a modified random walk on the half-plane with five transition probabilities. In contrast, the inverse trinomial arises as a random walk on the real line with three transition probabilities. The probability mass function (pmf) is expressible in terms of the Gauss hypergeometric function and this offers computational advantage due to its recurrence formula. The descending factorial moment is also obtained. The GIT contains twenty-two possible distributions in total. Special cases include the binomial, negative binomial, shifted negative binomial, shifted inverse binomial or, equivalently, lost-games, and shifted inverse trinomial distributions. A subclass GIT_3,1 is a particular member of Kemp’s class of convolution of pseudo-binomial variables and its properties such as reproductivity, formulation, pmf, moments, index of dispersion, and approximations are studied in detail. Compound or generalized (stopped sum) distributions provide inflated models. The inflated GIT_3,1 extends Minkova’s inflated-parameter binomial and negative binomial. A bivariate model which has the GIT as a marginal distribution is also proposed.     

First crossing time,overshoot and Appell–Hessenberg type functions

We consider a general insurance risk model with extended flexibility under which claims arrive according to a point process with independent increments, their amounts may have any joint distribution and the premium income is accumulated following any non-decreasing, possibly discontinuous, real valued function. Point processes with independent increments are in general non-stationary, allowing for an arbitrary (possibly discontinuous) claim arrival cumulative intensity function which is appealing for insurance applications. Under these general assumptions, we derive a closed form expression for the joint distribution of the time to ruin and the deficit at ruin, which is remarkable, since as we show, it involves a new interesting class of what we call Appell–Hessenberg type functions. The latter are shown to coincide with the classical Appell polynomials in the Poisson case and to yield a new class of the so called Appell–Hessenberg factorial polynomials in the case of negative binomial claim arrivals. Corollaries of our main result generalize previous ruin formulas e.g. those obtained for the case of stationary Poisson claim arrivals.     

Certain characterizations with linearity of regression in additive damage models

This paper investigates the characterizations of certain discrete distributions within the framework of a multivariate additive damage model. The univariate case for such a model appeared in an article by [2]. In this model a p-dimensional observation is subjected to damage according to a specified probability law represented by a joint survival distribution. Here, it is shown that the linearity of regression of the damaged part on the undamaged ones leads to the characterizations of the multivariate binomial, and multiple inverse hypergeometric distribution as survival distributions.     

Limiting behavior of statistics for some discrete distributions

We estimate the integral closeness of the Poisson, the Polya, and the negative Polya distributions to the normal. Some previous results on normal approximation to binomial and hypergeometric distributions are generalized.Translated from Statisticheskie Metody, pp. 104–113, 1980.     

Ruin problems for an autoregressive risk model with dependent rates of interest

In this paper, we consider a discrete insurance risk model in which the claims, the premiums and the rates of interest are assumed to have dependent autoregressive structures (AR(1)). We derive recursive and integral equations for expected discounted penalty function. By these equations, we obtain generalized Lundberg inequality for the infinite time severity of ruin and hence for the infinite time ruin probability, consider asymptotic formula for the finite time ruin probability when loss distributions have regularly varying tails, and study some probability properties of the duration of ruin.     

A Class of Ruin Probability Mo del with Dep endent Structure

In this paper, we study a class of ruin problems, in which premiums and claims are dependent. Under the assumption that premium income is a stochastic process, we raise the model that premiums and claims are dependent, give its numerical characteristics and the ruin probability of the individual risk model in the surplus process. In addition, we promote the number of insurance policies to a Poisson process with parameter λ, using martingale methods to obtain the upper bound of the ultimate ruin probability.     

On a correlated aggregate claims model with Poisson and Erlang risk processes

In this paper we consider a risk model with two dependent classes of insurance business. In this model the two claim number processes are correlated. Claim occurrences of both classes relate to Poisson and Erlang processes. We derive explicit expressions for the ultimate survival probabilities under the assumed model when the claim sizes are exponentially distributed. We also examine the asymptotic property of the ruin probability for this special risk process with general claim size distributions.     

一类离散双险种风险模型

本文推广了[1]的离散双险种风险模型,讨论了两类险种的索赔均为负二项随机序列的情形,得到了最终破产概率的Lundberg不等式以及一般表达式.     

Characterization of continuous distributions in terms of moments of extremal statistics

We give characterizations of continuous distributions in terms of moments of order statistics when the sample size is random. In particular, we characterize uniform, exponential, Pareto, and logistic distributions. Special cases of a random sample size (logarithmic series, geometrical, truncated binomial, truncated negative binomial, and truncated Poisson distribution) are also considered. Proceedings of the XVII Seminar on Stability Problems for Stochastic Models, Kazan, Russia, 1995, Part I.     

Multivariate Tweedie distributions and some related capital-at-risk analyses

We study a multivariate extension of the univariate exponential dispersion Tweedie family of distributions. The class, referred to as the multivariate Tweedie family (MTwF), on the one hand includes multivariate Poisson, gamma, inverse Gaussian, stable and compound Poisson distributions and on the other hand introduces a high variety of new dependent probabilistic models unstudied so far. We investigate various properties of MTwF and discuss its possible applications to financial risk management.     

相依索赔的二项风险模型的Gerber-shiu贴现罚函数

研究一类索赔时间相依的二项风险模型,根据索赔额的大小随机产生一副索赔.通过引入辅助模型,运用概率论的分析方法得到了任意初始值μ下的Gerber-Shiu贴现罚函数,并求得了初始值为0时最终破产概率的明确表达式.最后结合保险实务进行了举例.     

Generalized hypergeometric,digamma and trigamma distributions

Summary In this paper we introduce and study new probability distributions named “digamma” and “trigamma” defined on the set of all positive integers. They are obtained as limits of the zero-truncated Type B3 generalized hypergeometric distributions (inverse Pólya-Eggenberger or negative binomial beta distributions), and also by compounding the logarithmic series distributions. The family of digamma distributions has the logarithmic series as a limit and the trigamma as another limit. The trigamma distributions are very close to the zeta (Zipf) distributions. Thus, our new distributions are useful as substitutes of the logarithmic series when the observed frequency data have such a long tail that cannot be fitted by the latter distributions. In the beginning sections we summarize properties of the Type B3 generalized hypergeometric distributions. It is emphasized that the distributions are obtained by compounding a Poisson distribution by “gamma product-ratio” distributions.     

Finite Time Ruin Probabilities and Large Deviations for Generalized Compound Binomial Risk Models

In this paper, we extend the classical compound binomial risk model to the case where the premium income process is based on a Poisson process, and is no longer a linear function. For this more realistic risk model, Lundberg type limiting results for the finite time ruin probabilities are derived. Asymptotic behavior of the tail probabilities of the claim surplus process is also investigated.