Negative binomial distributions for point processes

Negative binomial point processes are defined for which all finite-dimensional distributions associated with disjoint bounded Borel sets are negative binomial in the usual sense. For these processes we study classical notions such as infinite divisibility, conditional distributions, Palm probabilities, convergence, etc. Negative binomial point processes appear to be of interest because they are mathematically tractable models which can be used in many situations. The general results throw some new light on some well-known special cases like the Polya process and the Yule process.     

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.     

Further modified forms of binomial and poisson distributions

Summary Some new type of modifications of binomial and Poisson distributions, are discussed. First, we consider Bernoulli trials of lengthn with success ratep up to time whenm times of successes occur, and then, changing the success rate to γp, we continue the remaining trial. The distribution of number of successes is called the modified binomial distribution. The Poisson limit (n tends to infinity andp tends to 0, keepingnp=λ) of the modified binomial is called the modified Poisson distribution. The probability functions of modified binomial and Poisson distributions are given (Section 1). A new concept of (m, γ)-modification is introduced and fundamental theorem which gives the relations between the factorial moments of any probability function and the factorial moments of its (m, γ)-modification, is presented. Then some lower order moments of the modified binomial and Poisson distributions are given explicitly (Section 2). The modified Poisson ofm=2 is fitted to the distribution of number of children for Japanese women in some age group. The fitting procedure is also presented (Section 3). Some historical sketch concerning the modification and generalization of binomial and Poisson distributions is given in Appendix. The Institute of Statistical Mathematics     

Uniformly distributed sums of bernoulli variables

Sums of independent, identically distributed (iid) binomial variates have binomial distributions; yet it is possible to construct a sequence of binomial distributions over {0, 1} for variatesX ₁,X ₂, ... such that all partial sumsY _i =X ₁ + ... +X _i have uniform distributions. The price to pay is to give up the iid condition. Requiring the property of only one sum does not alleviate the situation much.It is also possible to generate on a computerm × n-matrices, of 0–1 binomial variates with uniformly distributed row and column sums of all major submatrices, but only for smallm andn. Even a three-dimensional 2 × 2 × 2 array can have a similar property.Other target distributions than the rectangular are possible, but cumbersome. An example with smaller variance is given.The results were needed for simulating the performance of some Operations Research algorithms.Dedicated to Peter Naur on the occasion of his 60th birthday     

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.     

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.     

连续时间复合二项模型多维精算量的联合分布

连续时间复合二项模型是由文献首先提出的.作为离散时间复合二项模型的连续化版本,连续时间复合二项模型的极限形式即为经典风险模型.为了得到该模型多维精算量的联合分布,该文引入了一列上穿零点,推导出该列上穿零点所构成的缺陷(defective)更新序列的更新质量函数.利用此更新质量函数及余额过程的强马氏性可以得到破产概率和包含破产时间,破产前余额,破产严重程度,破产前最大盈余,破产到恢复的最大赤字,整个过程的最大赤字等多维精算量的联合分布.由此联合分布得到其1-骨架链—离散时间复合二项模型的对应的联合分布,最后给出在1-骨架链中索赔额服从指数分布时这一特殊情况下相应多维精算量的联合分布的明确表达式.     

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.     

Generation of Over-Dispersed and Under-Dispersed Binomial Variates

Abstract

This article proposes an algorithm for generating over-dispersed and under-dispersed binomial variates with specified mean and variance. The over-dispersed/under-dispersed distributions are derived from correlated binary variables with an underlying continuous multivariate distribution. Different multivariate distributions or different correlation matrices result in different over-dispersed (or under-dispersed) distributions. The over-dispersed binomial distributions that are generated from three different correlation matrices of a multivariate normal are compared with the beta-binomial distribution for various mean and over-dispersion parameters by quantile-quantile (Q-Q) plots. The two distributions appear to be similar. The under-dispersed binomial distribution is simulated to model an example data set that exhibits under-dispersed binomial variation.     

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.     

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.     

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。     

Bayesian inspection model with the negative binomial prior in the presence of inspection errors

One of the basic assumptions in Bayesian inspection models is that we have some prior knowledge about the number of defects in a certain product or software system. The prior knowledge can be often described as a probability distribution (e.g., Poisson distribution). In the paper, we propose three conditions that should be put forth as desirable properties for a prior probability distribution of the number of defects in the product. We review various prior probability distributions and test if they meet those conditions. The negative binomial distribution is found to be the only one that satisfies all the desirable conditions. With the negative binomial prior, we analyze the effects of various parameters on the Bayesian estimate of the number of undetected errors still remaining in the product.     

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.     

Binomial option pricing with nonidentically distributed returns and its implications

The argument of Cox, Ross, and Rubinstein for pricing options is generalized in the direction of using nonidentically distributed binomial returns as a model for the stock price process. It is found that the use of nonidentically distributed binomial returns, in the limit exhaust the class of infinitely divisible distributions. The pricing of these models are considered and it is shown that the model is a generalization of the Black-Scholes model. The use, however, of nonidentically distributed returns, it is shown, can lead to contradictions. Hence, it is argued, the models used for stock price behavior requires restrictions.     

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．     

Random sums of exchangeable variables and actuarial applications

In this paper we study the accumulated claim in some fixed time period, skipping the classical assumption of mutual independence between the variables involved. Two basic models are considered: Model 1 assumes that any pair of claims are equally correlated which means that the corresponding square-integrable sequence is exchangeable one. Model 2 states that the correlations between the adjacent claims are the same. Recurrence and explicit expressions for the joint probability generating function are derived and the impact of the dependence parameter (correlation coefficient) in both models is examined. The Markov binomial distribution is obtained as a particular case under assumptions of Model 2.     

Bayesian multiple change-point estimation with annealing stochastic approximation Monte Carlo

Bayesian multiple change-point models are built with data from normal, exponential, binomial and Poisson distributions with a truncated Poisson prior for the number of change-points and conjugate prior for the distributional parameters. We applied Annealing Stochastic Approximation Monte Carlo (ASAMC) for posterior probability calculations for the possible set of change-points. The proposed methods are studied in simulation and applied to temperature and the number of respiratory deaths in Seoul, South Korea.     

Bayesian and likelihood inference from equally weighted mixtures

Equally weighted mixture models are recommended for situations where it is required to draw precise finite sample inferences requiring population parameters, but where the population distribution is not constrained to belong to a simple parametric family. They lead to an alternative procedure to the Laird-DerSimonian maximum likelihood algorithm for unequally weighted mixture models. Their primary purpose lies in the facilitation of exact Bayesian computations via importance sampling. Under very general sampling and prior specifications, exact Bayesian computations can be based upon an application of importance sampling, referred to as Permutable Bayesian Marginalization (PBM). An importance function based upon a truncated multivariatet-distribution is proposed, which refers to a generalization of the maximum likelihood procedure. The estimation of discrete distributions, by binomial mixtures, and inference for survivor distributions, via mixtures of exponential or Weibull distributions, are considered. Equally weighted mixture models are also shown to lead to an alternative Gibbs sampling methodology to the Lavine-West approach.