On Mixed and Compound Mixed Poisson Distributions

Recursive formulae are derived for the evaluation of the t-th order cumulative distribution function and the t-th order tail probability of compound mixed Poisson distributions in the case where the derivative of the logarithm of the mixing density can be written as a ratio of polynomials. Also, some general results are derived for the evaluation of the t-th order moments of stop-loss transforms. The recursions can be applied for the exact evaluation of the probability function, distribution function, tail probability and stop-loss premium of compound mixed Poisson distributions and the corresponding mixed Poisson distributions. Several examples are also presented.     

On evaluation of the Delaporte distribution and related distributions

Abstract

In the present paper we develop recursive algorithms for evaluation of the Delaporte distribution, the compound Delaporte distribution, and convolutions of compound Delaporte distributions. Some asymptotic results are given. We discuss how the approach can sometimes be generalized to other classes of compound mixed Poisson distributions when the mixing distribution is a shifted infinitely divisible distribution.     

On the Probability of (Non-) Ruin in Infinite Time

In the context of the classical Poisson ruin model Gerber (1988a,b) and Shiu (1987, 1989) have obtained two formulae for the ruin and non ruin probabilities in infinite time. Here these two formulae are generalized to the case of an arbitrary premium process when all claims are integer-valued, as in Picard & Lefèvre (1997). Moreover, this generalization throws a new light on the two known formulae and it then leads very simply to a third new formula.     

On the Moments of the Time of Ruin with Applications to Phase-Type Claims

Abstract

We describe an approach to the evaluation of the moments of the time of ruin in the classical Poisson risk model. The methodology employed involves the expression of these moments in terms of linear combinations of convolutions involving compound negative binomial distributions. We then adapt the results for use in the practically important case involving phase-type claim size distributions. We present numerical examples to illuminate the influence of claim size variability on the moments of the time of ruin.     

Two-Sided Bounds for Tails of Compound Negative Binomial Distributions in the Exponential and Heavy-Tailed Cases

This paper derives two-sided bounds for tails of compound negative binomial distributions, both in the exponential and heavy-tailed cases. Two approaches are employed to derive the two-sided bounds in the case of exponential tails. One is the convolution technique, as in Willmot & Lin (1997). The other is based on an identity of compound negative binomial distributions; they can be represented as a compound Poisson distribution with a compound logarithmic distribution as the underlying claims distribution. This connection between the compound negative binomial, Poisson and logarithmic distributions results in two-sided bounds for the tails of the compound negative binomial distribution, which also generalize and improve a result of Willmot & Lin (1997). For the heavy-tailed case, we use the method developed by Cai & Garrido (1999b). In addition, we give two-sided bounds for stop-loss premiums of compound negative binomial distributions. Furthermore, we derive bounds for the stop-loss premiums of general compound distributions among the classes of HNBUE and HNWUE.     

Equilibrium compound distributions and stop-loss moments

A convolution representation is derived for the equilibrium or integrated tail distribution associated with a compound distribution. This result allows for the derivation of reliability properties of compound distributions, as well as an explicit analytic representation for the stop-loss premium, of interest in connection with insurance claims modelling. This result is extended to higher order equilibrium distributions, or equivalently to higher stop-loss moments. Special cases where the counting distribution is mixed Poisson or discrete phase-type are considered in some detail. An approach to handle more general counting distributions is also outlined.     

On a family of counting distributions and recursions for related compound distributions

Abstract

This paper considers a family of counting distributions whose densities satisfy certain second order difference equations. Recursions for the evaluation of related compound distributions are developed in the case of severity distributions which are concentrated on the non-negative integers. From these a characterization of the considered counting distributions is obtained, and it is shown that most of these are compound Poisson distributions.     

Two-Sided Bounds for Ruin Probabilities when the Adjustment Coefficient does not Exist

In this paper, we derive two-sided bounds for the ruin probability in the compound Poisson risk model when the adjustment coefficient of the individual claim size distribution does not exist. These bounds also apply directly to the tails of compound geometric distributions. The upper bound is tighter than that of Dickson (1994). The corresponding lower bound, which holds under the same conditions, is tighter than that of De Vylder and Goovaerts (1984). Even when the adjustment coefficient exists, the upper bound is, in some cases, tighter than Lundberg's bound. These bounds are applicable for any positive distribution function with a finite mean. Examples are given and numerical comparisons with asymptotic formulae for the ruin probability are also considered.     

Two-Sided Bounds for the Finite Time Probability of Ruin

Explicit, two-sided bounds are derived for the probability of ruin of an insurance company, whose premium income is represented by an arbitrary, increasing real function, the claims are dependent, integer valued r.v.s and their inter-occurrence times are exponentially, non-identically distributed. It is shown, that the two bounds coincide when the moments of the claims form a Poisson point process. An expression for the survival probability is further derived in this special case, assuming that the claims are integer valued, i.i.d. r.v.s. This expression is compared with a different formula, obtained recently by Picard & Lefevre (1997) in terms of generalized Appell polynomials. The particular case of constant rate premium income and non-zero initial capital is considered. A connection of the survival probability to multivariate B -splines is also established.     

操作风险损失的广义帕累托分布参数估计及其应用

极值理论表明大于某一阀值的样本服从广义帕累托分布,该结论在金融风险计量和保险精算中有着广泛的应用。然而,由于其参数没有可接受的估计方法,致使其应用受到限制。论文在推导出广义帕累托分布的条件矩的基础上,研究了基于操作风险损失的广义帕累托分布的参数估计问题。并且基于我国商业银行1994~2008年的操作风险损失数据对经济资本配置进行了算例分析。     

S -Convex Extrema,Taylor-Type Expansions and Stochastic Approximations

The present work studies s -convex orders using a remarkable probabilistic generalization of Taylor's theorem obtained by Massey & Whitt (1993) and further discussed by Lin (1994). We propose two methods for approximating a given risk with known first moments by means of s -convex extremal distributions. The goodness of those approximations is explored using stop-loss distances. Several applications show the interest of this approach in actuarial sciences.     

Some results on moments and cumulants

Abstract

In the present paper we discuss various results related to moments and cumulants of probability distributions and approximations to probability distributions. As the approximations are not necessarily probability distributions themselves, we shall apply the concept of moments and cumulants to more general functions. Recursions are deduced for moments and cumulants of functions in the form R_k [a, b] as defined by Dhaene & Sundt (1996). We deduce a simple relation between the De Pril transform and the cumulants of a function. This relation is applied to some classes of approximations to probability distributions, in particular the approximations of Hipp and De Pril.     

A note on the connection between factorial series distribution and zero-truncated power series distribution

Summary

The main purpose of this note is to call attention to the close relationship between the problem of estimating the parameter of a zero-truncated power series distribution and the problem of estimating the parameter of an associated factorial series distribution. We also extend a known recurrence property for the Poisson case to the binomial and negative binomial cases.     

Moment generating functions of compound renewal sums with discounted claims

Léveillé & Garrido (2001a, 2001b) have obtained recursive formulas for the moments of compound renewal sums with discounted claims, which incorporate both, Andersen's (1957) generalization of the classical risk model, where the claim number process is an ordinary renewal process, and Taylor's (1979), where the joint effect of the claims cost inflation and investment income on a compound Poisson risk process is considered.

In this paper, assuming certain regularity conditions, we improve the preceding results by examining more deeply the asymptotic and finite time moment generating functions of the discounted aggregate claims process. Examples are given for claim inter-arrival times and claim severity following phase-type distributions, such as the Erlang case.     

On moments of order statistics from the pareto distribution

Abstract

The evaluation of multiple integrals which occur in order statistics distribution theory is involved due to the fact that the integration is to be carried on over an ordered range of variables of integration. This difficulty is sometimes completely obviated by transforming the ordered variates to the unordered ones. Several such transformations are available in the Theory of Multiple Integrals. In previous papers [2, 3] the author used one such transformation, and gave alternative simplified proofs of several known results in the distribution theory of order statistics from the exponential and the power function distributions. In this paper we use such a known transformation to derive moments (and distributions if necessary) of order statistics from the Pareto distribution. Malik [4] has derived moments of order statistics from this distribution without the transformation of the ordered variates to the unordered ones. The process of direct integration used by Malik becomes complicated for dealing with the moments of more than two ordered variates. Further, the method which we use here is unformly applicable to derive the moments or the distributions of one or more ordered variables, and gives the distributions and moments without any complicated steps in integration. The transformation used by us considerably simplifies the manipulations necessary for the derivation of moments or the Mellin transforms, and thus we hope that our paper would at least be of Pedagogical interest.     

A fitting procedure for some generalized poisson distributions

Abstract

Many of the contagious distributions considered in the biological sciences are members of the generalized Poisson family. Four distributions which belong to this family and have been used frequently are the Negative Binomial (cf. Bliss [2]), Neyman Type A (cf. Beall and Rescia [1]), Poisson Binomial (cf. McGuire et al. [10]) and the generalized Polya-Aeppli (cf. Skellam [14]).     

Moment relations for some discrete distributions

Summary

Recursion relations for moments not involving differentiation with respect to the parameter of the distribution are given for the Poisson distribution in Philipson (1963) and those for the logarithmic series distribution in Patil, Kamat and Wani (1964) and Patil and Wani (1965). Similar results are obtained here for the negative binomial distribution. The relations referred to above are obtained as corollaries.     

Remarks on “A note on compound generalized distributions”

Abstract

1. INTRODUCTION

In [1] the authors derive recursion formulae for computing total claim probabilities for a general class of modified power series distributions. Such formulae provide an important tool for computing total claim size probabilities in risk-theory. As it turns out, two of their recursions (Theorem 3.2 and Theorem 3.4) need modifications. Unfortunately, these modifications have the effect that the recursions break down. In the following, we will state the modified theorems and show how these obstacles can be overcome.     

Semiparametric Estimation for Non-Ruin Probabilities

We consider the classical risk model with unknown claim size distribution F and unknown Poisson arrival rate u . Given a sample of claims from F and a sample of interarrival times for these claims, we construct an estimator for the function Z ( u ), which gives the probability of non-ruin in that model for initial surplus u . We obtain strong consistency and asymptotic normality for that estimator for a large class of claim distributions F . Confidence bounds for Z ( u ) based on the bootstrap are also given and illustrated by some numerical examples.     

On Tail Value-at-Risk for sums of non-independent random variables with a generalized Pareto distribution

Recently in actuarial literature several authors have derived lower and upper bounds in the sense of convex order for sums of random variables with given marginal distributions and unknown dependency structure. In this paper, we derive convex bounds for sums of non-independent and identically distributed random variables when marginal distributions are mixture models. In particular, we examine some well-known risk measures and we find approximations for Tail Value-at-Risk of the sums considered when marginal distributions are generalized Pareto distributions. By numerical examples we illustrate the goodness of the presented approximations.