On some compound distributions with Borel summands

The generalized Poisson distribution is well known to be a compound Poisson distribution with Borel summands. As a generalization we present closed formulas for compound Bartlett and Delaporte distributions with Borel summands and a recursive structure for certain compound shifted Delaporte mixtures with Borel summands. Our models are introduced in an actuarial context as claim number distributions and are derived only with probabilistic arguments and elementary combinatorial identities. In the actuarial context related compound distributions are of importance as models for the total size of insurance claims for which we present simple recursion formulas of Panjer type.     

A class of bivariate negative binomial distributions with different index parameters in the marginals

In this paper, we consider a new class of bivariate negative binomial distributions having marginal distributions with different index parameters. This feature is useful in statistical modelling and simulation studies, where different marginal distributions and a specified correlation are required. This feature also makes it more flexible than the existing bivariate generalizations of the negative binomial distribution, which have a common index parameter in the marginal distributions. Various interesting properties, such as canonical expansions and quadrant dependence, are obtained. Potential application of the proposed class of bivariate negative binomial distributions, as a bivariate mixed Poisson distribution, and computer generation of samples are examined. Numerical examples as well as goodness-of-fit to simulated and real data are also given here in order to illustrate the application of this family of bivariate negative binomial distributions.     

离散型广义非线性纵向数据模型中偏离名义离差的检验及其功效模拟

离散型广义非线性模型包括Poisson,二项,负二项模型.本文讨论离散型广义非线性纵向数据模型中偏离名义离差的检验问题,得到了检验的score统计量,并利用MonteCarlo方法研究了检验统计量的性质.最后,利用杀虫剂数据说明了检验方法的应用.     

Multivariate distributions of order κ

Three multivariate distributions of order κ are introduced and studied. A multivariate negative binomial distribution of order κ is derived first, by means of an urn scheme, and two limiting cases of it are obtained next. They are, respectively, a multivariate Poisson distribution of order κ and a multivariate logarithmic series distribution of the same order. The probability generating functions, means variances and covariances of these distributions are obtained, and some further genesis schemes of them and interrelationships among them are also established. The present paper extends to the multivariate case the work of Philippou (1987) on multiparameter distributions of order κ. At the same time, several results of Aki (1985) on extended distributions of order κ are also generalized to the multivariate case.     

Decompounding random sums: a nonparametric approach

A compound distribution is the distribution of a random sum, which consists of a random number N of independent identically distributed summands, independent of N. Buchmann and Grübel (Ann Stat 31:1054–1074, 2003) considered decompounding a compound Poisson distribution, i.e. given observations on a random sum when N has a Poisson distribution, they constructed a nonparametric plug-in estimator of the underlying summand distribution. This approach is extended here to that of general (but known) distributions for N. Asymptotic normality of the proposed estimator is established, and bootstrap methods are used to provide confidence bounds. Finally, practical implementation is discussed, and tested on simulated data. In particular we show how recursion formulae can be inverted for the Panjer class in general, as well as for an example drawn from the Willmot class.     

关于溢额损失再保险的条件分布的递推方程

在索赔数目服从Poisson分布、二项分布或负二项分布,以及索赔额分布的密度函数连续且有界的条件下,研究了溢额损失再保险条款的总体损失分布的条件递推方程．在再保险人或分出人的索赔数目给定的条件下,得到了再保险人以及分出人的总赔付额分布的递推方程．     

广义泊松回归模型的推广及其在医疗保险中应用

计数数据往往存在过离散(over-dispersed)即方差大于均值特征,若利用传统的泊松回归模型拟合数据往往会导致其参数的标准误差被低估,显著性水平被高估的错误结论。负二项回归模型、广义泊松回归模型通常被用来处理过离散特征数据。本文以两类广义泊松回归模型GP-1和GP-2模型为基础,将其推广为更为一般的GP-P形式,其中P为参数。此时,P=1或P=2,GP-P模型就退化为GP-1和GP-2模型。文中最后利用此类推广的GP-P模型处理了一组医疗保险数据,并与泊松回归模型、负二项回归模型拟合结果进行了比较。结果表明,推广后的GP-P模型的拟合效果更优。     

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.     

An expansion in the exponent for compound binomial approximations

The purpose of this paper is two-fold. First, we introduce a new asymptotic expansion in the exponent for the compound binomial approximation of the generalized Poisson binomial distribution. The dependence of its accuracy on the symmetry and shifting of distributions is investigated. Second, for compound binomial and compound Poisson distributions, we present new smoothness estimates, some of which contain explicit constants. Finally, the ideas used in this paper enable us to prove new precise bounds in the compound Poisson approximation. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 1, pp. 67–110, January–March, 2006.     

On generalized binomial and multinomial distributions and their relation to generalized Poisson distributions

Summary The binomial and multinomial distributions are, probably, the best known distributions because of their vast number of applications. The present paper examines some generalizations of these distributions with many practical applications. Properties of these generalizations are studied and models giving rise to them are developed. Finally, their relationship to generalized Poisson distributions is examined and limiting cases are given.     

Properties of the extended distributions of order κ

Relationships among the extended negative binomial, the extended Poisson and the extended logarithmic series distributions of order κ are discussed. The results are extentions of those for the usual distributions.     

Numerical algorithms for Panjer recursion by applying Bernstein approximation

In actuarial science, Panjer recursion (1981) is used in insurance to compute the loss distribution of the compound risk models. When the severity distribution is continuous with density function, numerical calculation for the compound distribution by applying Panjer recursion will involve an approximation of the integration. In order to simplify the numerical algorithms, we apply Bernstein approximation for the continuous severity distribution function and obtain approximated recursive equations, which are used for computing the approximated values of the compound distribution. The theoretical error bound for the approximation is also obtained. Numerical results show that our algorithm provides reliable results.     

Fitting the extended poisson process model to grouped binary data

Summary  Extended Poisson process modelling allows the construction of a broad class of distributions, including distributions over-dispersed or under-dispersed relative to the binomial distribution, with the binomial distribution being a special case. In this paper an iteratively re-weighted least squares algorithm for fitting such generalised binomial distributions is presented, and is illustrated with an example.     

An elementary proof of the Adelson—Panjer recursion formula

We present a new proof of the Adelson—Panjer recursion formula for computing discrete Compound Poisson densities. It uses only elementary probability theory and elementary arithmetic.     

Complete monotonicity of some entropies

It is well-known that the Shannon entropies of some parameterized probability distributions are concave functions with respect to the parameter. In this paper we consider a family of such distributions (including the binomial, Poisson, and negative binomial distributions) and investigate their Shannon, Rényi, and Tsallis entropies with respect to complete monotonicity.     

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.     

Distributional analysis of a generalization of the Polya process

A nonhomogeneous birth process generalizing the Polya process is analyzed, and the distribution of the transition probabilities is shown to be the convolution of a negative binomial distribution and a compound Poisson distribution, whose secondary distribution is a mixture of zero-truncated geometric distributions. A simplified form of the secondary distribution is obtained when the transition intensities have a particular structure, and may sometimes be expressed in terms of Stirling numbers and special functions such as the incomplete gamma function, the incomplete beta function, and the exponential integral. Conditions under which the compound Poisson form of the marginal distributions may be improved to a geometric mixture are also given.     

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.     

Strategies for computation of compound distributions with two-sided severities

The recursive scheme proposed by Panjer for the computation of discrete compound distributions whose counting distributions are Binomial, Poisson, or Pascal provides an extremely efficient alternative to an important actuarial problem where brute-force convolution methods or moment-based approximations are unsatisfactory. However, the scheme fails in the important case where the counts are Pascal and the severities are two-sided. As research is currently underway on this problem, the authors intend to present computational results on the following approaches: (1) iterative use of Panjer's formula: (2) approximation by splitting of positive and negative components: (3) use of analytically-derived tail approximations.     

Risk aggregation in multivariate dependent Pareto distributions

In this paper we obtain closed expressions for the probability distribution function of aggregated risks with multivariate dependent Pareto distributions. We work with the dependent multivariate Pareto type II proposed by Arnold (1983, 2015), which is widely used in insurance and risk analysis. We begin with an individual risk model, where the probability density function corresponds to a second kind beta distribution, obtaining the VaR, TVaR and several other tail risk measures. Then, we consider a collective risk model based on dependence, where several general properties are studied. We study in detail some relevant collective models with Poisson, negative binomial and logarithmic distributions as primary distributions. In the collective Pareto–Poisson model, the probability density function is a function of the Kummer confluent hypergeometric function, and the density of the Pareto–negative binomial is a function of the Gauss hypergeometric function. Using data based on one-year vehicle insurance policies taken out in 2004–2005 (Jong and Heller, 2008) we conclude that our collective dependent models outperform other collective models considered in the actuarial literature in terms of AIC and CAIC statistics.