On discrete distributions of orderk

Summary This paper gives some results on calculation of probabilities and moments of the discrete distributions of orderk. Further, a new distribution of orderk, which is called the logarithmic series distribution of orderk, is investigated. Finally, we discuss the meaning of theorder of the distributions. The Institute of Statistical Mathematics     

Discrete distributions of orderk on a binary sequence

Summary This paper considers discrete distributions of orderk based on a binary sequence which is defined as an extension of independent trials with a constant success probability and is more practical than the independent trials. Some results on calculation of probabilities and characteristics of the distributions are obtained as well as their formal expressions. Examples and an application are also given. The Institute of Statistical Mathematics     

On a family of aggregate claims distributions

This paper gives a family of aggregate claims distributions using an integral equation representation. The Kernel of the integral equation is chosen so that the compound distributions such as Poisson and binomial are members of the same family. Furthermore, the aggregate claims distribution can be determined by the mean and variance of the number of claims.     

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.     

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.     

A space of generalized distributions

In this paper we use a duality method to introduce a new space of generalized distributions. This method is exactly the same introduced by Schwartz for the distribution theory. Our space of generalized distributions contains all the Schwartz distributions and all the multipole series of physicists and is, in a certain sense, the smallest space containing all these series. To The Memory of Laurent Schwartz     

On characterization of power series distributions by a marginal distribution and a regression function

The conditional distribution of Y given X=x, where X and Y are non-negative integer-valued random variables, is characterized in terms of the regression function of X on Y and the marginal distribution of X which is assumed to be of a power series form. Characterizations are given for a binomial conditional distribution when X follows a Poisson, binomial or negative binomial, for a hypergeometric conditional distribution when X is binomial and for a negative hypergeometric conditional distribution when X follows a negative binomial.     

On a new system of discrete distributions and characterizations of several discrete distributions by equality of distributions

This paper deals with a new system of discrete distributions. It also gives several characterizations of the Waring (and hence the Yule) distribution (and its truncated versions), the super-Poisson, the discrete uniform and other discrete distributions by using this system and other such systems existing in the literature, and linear regression. Continuous analogues of the above results are also briefly discussed.     

Log-concavity of compound distributions with applications in stochastic optimization

Compound distributions come up in many applications (telecommunication, hydrology, insurance, etc.), where some of the typical problems are of optimization type. The log-concavity property is paramount in these respects to ensure convexity. In this paper, we prove the log-concavity of some compound Poisson and other compound distributions.     

Asymptotically minimum variance unbiased estimation for a class of power series distributions

Summary The problem of finding an asymptotically minimum variance unbiased estimator (A.M.V.U.E.) for the parameter of certain truncated power series distributions, is discussed, when the generating function of their coefficients are i) polynomials of binomial type ii) generalized ascending factorials iii) polynomials with coefficients the well known Eulerian numbers.     

On zeros of discrete orthogonal polynomials

We exploit difference equations to establish sharp inequalities on the extreme zeros of the classical discrete orthogonal polynomials, Charlier, Krawtchouk, Meixner and Hahn. We also provide lower bounds on the minimal distance between their consecutive zeros.     

On the Lagrangian Katz family of distributions as a claim frequency model

The Panjer (Katz) family of distributions is defined by a particular first-order recursion which is built on the basis of two parameters. It is known to characterize the Poisson, negative binomial and binomial distributions. In insurance, its main usefulness is to yield a simple recursive algorithm for the aggregate claims distribution. The present paper is concerned with the more general Lagrangian Katz family of distributions. That family satisfies an extended recursion which now depends on three parameters. To begin with, this recursion is derived through a certain first-crossing problem and two applications in risk theory are described. The distributions covered by the recursion are then identified as the generalized Poisson, generalized negative binomial and binomial distributions. A few other properties of the family are pointed out, including the index of dispersion, an extended Panjer algorithm for compound sums and the asymptotic tail behaviour. Finally, the relevance of the family is illustrated with several data sets on the frequency of car accidents.     

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。     

On strong unimodality of multivariate discrete distributions

A discrete function f defined on Zⁿ is said to be logconcave if for , , . A more restrictive notion is strong unimodality. Following Barndorff-Nielsen [O. Barndorff-Nielsen, Unimodality and exponential families, Commun. Statist. 1 (1973) 189-216] a discrete function is called strongly unimodal if there exists a convex function such that  if . In this paper sufficient conditions that ensure the strong unimodality of a multivariate discrete distribution, are given. Examples of strongly unimodal multivariate discrete distributions are presented.     

Characterizations of mixtures of discrete distributions by a regression point

Given that the conditional distribution p_s(y|x) of Y, given X = x is an x-fold convolution of a nonnegative integer-valued r.v. ξ for every s= P[ξ = 0] > 0, the distribution of X, hence also of Y, is characterized by the regression point m(0) = E[X|Y = 0]. An infinite variety of generalized distributions (of Y) can be characterized by arbitrarily varying the distribution of X.     

Asymptotic Eulerian expansions for binomial and negative binomial reciprocals

Asymptotic expansions of any order for expectations of inverses of random variables with positive binomial and negative binomial distributions are obtained in terms of the Eulerian polynomials. The paper extends and improves upon an expansion due to David and Johnson (1956-7).

     

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．     

Approximation of aggregate claims distributions by compound poisson distributions

New error bounds are derived for the approximation of aggregate claims distributions by compound Poisson distributions. These approximations can be recommended in most cases in which the normal approximation fails.     

Estimation of parameters in the discrete distributions of order k

This paper considers estimating parameters in the discrete distributions of order k such as the binomial, the geometric, the Poisson and the logarithmic series distributions of order k. It is discussed how to calculate maximum likelihood estimates of parameters of the distributions based on independent observations. Further, asymptotic properties of estimators by the method of moments are investigated. In some cases, it is found that the values of asymptotic efficiency of the moment estimators are surprisingly close to one.