Lower-Bound Estimates for Poisson-Type Approximations

Let S = w ₁ S ₁ + w ₂ S ₂ + ⋯ + w _N S _N . Here S _j is a sum of identically distributed random variables with weight w _j > 0. We consider the cases where S _j is a sum of independent random variables, the sum of independent lattice variables, or has the Markov binomial distribution. Apart from the general case, we investigate the case of symmetric random variables. Distribution of S is approximated by a compound Poisson distribution, by a second-order asymptotic expansion, and by a signed exponential measure. Lower bounds for the accuracy of approximations in uniform metric are established. __________ Translated from Lietuvos Matematikos Rinkinys, Vol. 45, No. 4, pp. 501–524, October–December, 2005.     

A Simple Method for Computing the Observed Information Matrix When Using the EM Algorithm with Categorical Data

Abstract

A simple matrix formula is given for the observed information matrix when the EM algorithm is applied to categorical data with missing values. The formula requires only the design matrices, a matrix linking the complete and incomplete data, and a few simple derivatives. It can be easily programmed using a computer language with operators for matrix multiplication, element-by-element multiplication and division, matrix concatenation, and creation of diagonal and block diagonal arrays. The formula is applicable whenever the incomplete data can be expressed as a linear function of the complete data, such as when the observed counts represent the sum of latent classes, a supplemental margin, or the number censored. In addition, the formula applies to a wide variety of models for categorical data, including those with linear, logistic, and log-linear components. Examples include a linear model for genetics, a log-linear model for two variables and nonignorable nonresponse, the product of a log-linear model for two variables and a logit model for nonignorable nonresponse, a latent class model for the results of two diagnostic tests, and a product of linear models under double sampling.     

Reliability and non-reliability studies of Poisson variables in series and parallel systems

In this paper, we have derived the distribution of the minimum and maximum of two independent Poisson random variables. A useful procedure for computing the probabilities is given and a total of four numerical examples are presented. Of these four examples, the first two are on the generated data and the other two are on the Champion League Soccer data in order to illustrate the model which is considered here. The hazard rate and the reversed hazard rate, of the minimum and maximum of two independent discrete random variables, are also obtained and their monotonicity is investigated. The results for the Poisson-distributed variables are obtained as special cases.     

Poisson on the poisson distribution

A translation of the totality of Poisson's own 1837 discussion of the Poisson distribution is presented, and its relation to earlier work of De Moivre is briefly noted.     

渐近正态随机变量函数的极限分布

基于渐近正态随机变量,导出随机变量函数极限分布的两个一般性理论结果.作为应用,证明了渐近正态随机变量一系列具体函数的极限分布,其中包括泊松随机变量平方根的渐近正态性,以及随机变量部分和在正则化常数是随机变量情况下的渐近正态性.     

Log-concavity and the maximum entropy property of the Poisson distribution

We prove that the Poisson distribution maximises entropy in the class of ultra log–concave distributions, extending a result of Harremoës. The proof uses ideas concerning log-concavity, and a semigroup action involving adding Poisson variables and thinning. We go on to show that the entropy is a concave function along this semigroup.     

直观实用的泊松图

本文介绍一种用来诊断数据是否来自泊松总体的方法—泊松图，并通过实例分析，说明诊断过程。     

整值随机变量序列的一类强律

本文引进似然比作为整值随机变量序列相对于服从Poisson分布的独立随机变量序列的偏差的一种度量，并通过限制似然比给出了样水空间的某种子集．在这种子集上得到了一类用不等式表示的强律，独立随机变量序列的一类强律是其特例．     

Accuracy of approximation in the Poisson theorem in terms of the χ<Superscript>2</Superscript>-distance

We study the limit behavior of the χ²-distance between the distributions of the nth partial sum of independent not necessarily identically distributed Bernoulli random variables and the accompanying Poisson law. As a consequence in the i.i.d. case we make the multiplicative constant preciser in the available upper bound for the rate of convergence in the Poisson limit theorem.     

Poisson过程分解问题的一个注记

在研究Poisson过程分解问题时,现有文献的证明往往令人费解,本文主要运用极限理论,给出了一个简明易懂的证明.     

A COMPOUND POISSON MODEL FOR LEARNING DISCRETE BAYESIAN NETWORKS

We introduce here the concept of Bayesian networks, in compound Poisson model, which provides a graphical modeling framework that encodes the joint probability distribution for a set of random variables within a directed acyclic graph. We suggest an approach proposal which offers a new mixed implicit estimator. We show that the implicit approach applied in compound Poisson model is very attractive for its ability to understand data and does not require any prior information. A comparative study between learned estimates given by implicit and by standard Bayesian approaches is established. Under some conditions and based on minimal squared error calculations, we show that the mixed implicit estimator is better than the standard Bayesian and the maximum likelihood estimators. We illustrate our approach by considering a simulation study in the context of mobile communication networks.     

Large-deviation probabilities for maxima of sums of subexponential random variables with application to finite-time ruin probabilities

We establish an asymptotic relation for the large-deviation probabilities of the maxima of sums of subexponential random variables centered by multiples of order statistics of i.i.d.standard uniform random variables.This extends a corresponding result of Korshunov.As an application,we generalize a result of Tang,the uniform asymptotic estimate for the finite-time ruin probability,to the whole strongly subexponential class.     

A shorter proof of Kanter’s Bessel function concentration bound

We give a shorter proof of Kanter’s (J. Multivariate Anal. 6, 222–236, 1976) sharp Bessel function bound for concentrations of sums of independent symmetric random vectors. We provide sharp upper bounds for the sum of modified Bessel functions I₀(x) + I₁(x), which might be of independent interest. Corollaries improve concentration or smoothness bounds for sums of independent random variables due to Čekanavičius & Roos (Lith. Math. J. 46, 54–91, 2006); Roos (Bernoulli, 11, 533–557, 2005), Barbour & Xia (ESAIM Probab. Stat. 3, 131–150, 1999), and Le Cam (Asymptotic Methods in Statistical Decision Theory. Springer, Berlin Heidelberg New York, 1986).      

带有相依重尾冲击的Poisson噪音过程尾的一致渐近性质

本文考虑了带有某种相依重尾冲击的Poisson噪音过程尾的一致渐近性质.当冲击是二元上尾渐近独立的非负随机变量具有长尾和控制变化尾分布且噪音函数具有正的上下界时,得到了过程尾概率的一致渐近公式.进而,当冲击具有连续的一致变化尾分布时,去除了噪音函数具有正的下界的限制.对于噪音函数不一定具有正的上界的情形,当冲击具有两两负象限相依结构时,也得到了一致渐近性结果.     

Set partitions and moments of random variables

It is well known that the sequence of Bell numbers (Bn)_n?0 (Bn being the number of partitions of the set [n]) is the sequence of moments of a mean 1 Poisson random variable τ (a fact expressed in the Dobiński formula), and the shifted sequence (Bn+1)_n?0 is the sequence of moments of 1+τ. In this paper, we generalize these results by showing that both and (where is the number of m-partitions of [n], as they are defined in the paper) are moment sequences of certain random variables. Moreover, such sequences also are sequences of falling factorial moments of related random variables. Similar results are obtained when is replaced by the number of ordered m-partitions of [n]. In all cases, the respective random variables are constructed from sequences of independent standard Poisson processes.     

Large deviation results for generalized compound negative binomial risk models

In this paper we extend and improve some results of the large deviation for random sums of random variables. Let {Xn;n 〉 1} be a sequence of non-negative, independent and identically distributed random variables with common heavy-tailed distribution function F and finite mean μ ∈R^＋, {N（n）; n ≥0} be a sequence of negative binomial distributed random variables with a parameter p C （0, 1）, n ≥ 0, let {M（n）; n ≥ 0} be a Poisson process with intensity λ 〉 0. Suppose {N（n）; n ≥ 0}, {Xn; n≥1} and {M（n）; n ≥ 0} are mutually independent. Write S（n） =N（n）∑i=1 Xi-cM（n）.Under the assumption F ∈ C, we prove some large deviation results. These results can be applied to certain problems in insurance and finance.     

On Sums of Products of Bernoulli Variables and Random Permutations

Let {X _k } _k1 be independent Bernoulli random variables with parameters p _k . We study the distribution of the number or runs of length 2: that is . Let S=lim _n S _n . For the particular case p _k =1/(k+B), B being given, we show that the distribution of S is a Beta mixture of Poisson distributions. When B=0 this is a Poisson(1) distribution. For the particular case p _k =p for all k we obtain the generating function of S _n and the limiting distribution of S _n for .     

Poisson Distribution in the Polynomial Semigroup

We consider the asymptotic behavior of the distributions of arithmetic functions in polynomial semigroups.__________Published in Lietuvos Matematikos Rinkinys, Vol. 44, No. 4, pp. 429–442, October–December, 2004.     

复合泊松过程的可加性

对复合泊松分布可加性的研究在许多的文献中都可以看到,本文首先应用特征函数的方法证明了复合泊松分布的可加性.以此为基础,结合对随机过程相关性质的讨论,证明了复合泊松过程也具有与复合泊松分布可加性相似的,某种意义上的可加性性质.     

Characterizations of the Poisson process as a renewal process via two conditional moments

Given two independent positive random variables, under some minor conditions, it is known that fromE(X^rX+Y)=a(X+Y)^r andE(X^sX+Y)=b(X+Y)^s, for certain pairs ofr ands, wherea andb are two constants, we can characterizeX andY to have gamma distributions. Inspired by this, in this article we will characterize the Poisson process among the class of renewal processes via two conditional moments. More precisely, let {A(t), t0} be a renewal process, with {S _k, k1} the sequence of arrival times, andF the common distribution function of the inter-arrival times. We prove that for some fixedn andk, kn, ifE(S _k ^r A(t)=n)=at^r andE(S _k ^s A(t)=n)=bt^s, for certain pairs ofr ands, wherea andb are independent oft, then {A(t), t0} has to be a Poisson process. We also give some corresponding results about characterizingFto be geometric whenF is discrete.Support for this research was provided in part by the National Science Council of the Republic of China, Grant No. NSC 81-0208-M110-06.