MULTIVARIATE EXTREME VALUE DISTRIBUTION AND ITS FISHER INFORMATION MATRIX

ＭＵＬＴＩＶＡＲＩＡＴＥＥＸＴＲＥＭＥＶＡＬＵＥＤＩＳＴＲＩＢＵＴＩＯＮＡＮＤＩＴＳＦＩＳＨＥＲＩＮＦＯＲＭＡＴＩＯＮＭＡＴＲＩＸ￥ＳＨＩＤＡＯＪＩ（史道济）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＴｉａｎｊｉｎＵｎｉｖｅｒｓｉｔｙ，Ｔｉａ...     

Type II bivariate Pólya–Aeppli distribution

In this paper, we introduce the Type II bivariate Pólya–Aeppli distribution as a compound Poisson distribution with bivariate geometric compounding distribution. The probability mass function, recursion formulas, conditional distributions and some other properties are then derived for this distribution.     

A Two-Parameter Distribution Obtained by Compounding the Generalized Exponential and Exponential Distributions

We introduce a new two-parameter lifetime distribution obtained by compounding the generalized exponential and exponential distributions. We assume that the shape parameter of the generalized exponential distribution is a random variable having the exponential distribution. The shapes of the density and hazard rate functions are derived. The model parameters are estimated by maximum likelihood, and an application of the proposed distribution is presented.

     

A Skew Laplace Distribution on Integers

We propose a discrete version of the skew Laplace distribution. In contrast with the discrete normal distribution, here closed form expressions are available for the probability density function, the distribution function, the characteristic function, the mean, and the variance. We show that this distribution on integers shares many properties of the skew Laplace distribution on the real line, including unimodality, infinite divisibility, closure properties with respect to geometric compounding, and a maximum entropy property. We also discuss statistical issues of estimation under this model.     

Markovian bulk-arrival and bulk-service queues with state-dependent control

We study a modified Markovian bulk-arrival and bulk-service queue incorporating state-dependent control. The stopped bulk-arrival and bulk-service queue is first investigated and the relationship with our queueing model is examined and exploited. Equilibrium behaviour is studied and the probability generating function of the equilibrium distribution is obtained. Queue length behaviour is also examined and the Laplace transform of the queue length distribution is presented. The important questions regarding hitting time and busy period distributions are answered in detail and the Laplace transforms of these distributions are presented. Further properties including expectations of hitting times and busy period are also explored.     

An efficient third-moment saddlepoint approximation for probabilistic uncertainty analysis and reliability evaluation of structures

This paper presents an efficient third-moment saddlepoint approximation approach for probabilistic uncertainty analysis and reliability evaluation of random structures. By constructing a concise cumulant generating function (CGF) for the state variable according to its first three statistical moments, approximate probability density function and cumulative distribution function of the state variable, which may possess any types of distribution, are obtained analytically by using saddlepoint approximation technique. A convenient generalized procedure for structural reliability analysis is then presented. In the procedure, the simplicity of general moment matching method and the accuracy of saddlepoint approximation technique are integrated effectively. The main difference of the presented method from existing moment methods is that the presented method may provide more detailed information about the distribution of the state variable. The main difference of the presented method from existing saddlepoint approximation techniques is that it does not strictly require the existence of the CGFs of input random variables. With the advantages, the presented method is more convenient and can be used for reliability evaluation of uncertain structures where the concrete probability distributions of input random variables are known or unknown. It is illustrated and examined by five representative examples that the presented method is effective and feasible.     

Start-Up Demonstration Tests with Rejection of Units upon Observing d Failures

The probability generating function of number of trials for the start-up demonstration test with rejection upon d failures is derived. The exact distribution of the number of trials is obtained. Some recurrence relations for the probabilities are also established. The average length of the test is derived. Some illustrative examples are finally presented.     

Start-Up Demonstration Tests Under Markov Dependence Model with Corrective Actions

A general probability model for a start-up demonstration test is studied. The joint probability generating function of some random variables appearing in the Markov dependence model of the start-up demonstration test with corrective actions is derived by the method of probability generating function. By using the probability generating function, several characteristics relating to the distribution are obtained.     

A compound of the generalized negative binomial distribution with the generalized beta distribution

This paper presents a compound of the generalized negative binomial distribution with the generalized beta distribution. In the introductory part of the paper, we provide a chronological overview of recent developments in the compounding of distributions, including the Polish results. Then, in addition to presenting the probability function of the compound generalized negative binomial-generalized beta distribution, we present special cases as well as factorial and crude moments of some compound distributions.     

Mathematical crystal growth II

A method is given for computing the generating function for a sequence of polynomials representing a growing crystal. The younger twin of this paper presented a non-constructive proof that the generating function is rational.     

An alternative method for computing mean and covariance matrix of some multivariate distributions

Computing the mean and covariance matrix of some multivariate distributions, in particular, multivariate normal distribution and Wishart distribution are considered in this article. It involves a matrix transformation of the normal random vector into a random vector whose components are independent normal random variables, and then integrating univariate integrals for computing the mean and covariance matrix of a multivariate normal distribution. Moment generating function technique is used for computing the mean and covariances between the elements of a Wishart matrix. In this article, an alternative method that uses matrix differentiation and differentiation of the determinant of a matrix is presented. This method does not involve any integration.     

任意随机序列关于二重非齐次马氏链的随机和的一类随机选择系统的随机逼近定理

本文引进任意随机变量序列随机极限对数似然比的概念,通过测度$\pr$下任意相依随机序列联合分布与测度$\qr$下二重非齐次马氏分布相比较,利用母函数与尾概率母函数工具研究任意受控随机序列之随机和在随机选择系统中的一类随机逼近定理.     

A continuous time markov-renewal replacement model for manpower systems

A continuous time Markov-renewal model is presented that generalizes the classical Young and Almond model for manpower systems with given size. The construction is based on the associated Markov-renewal replacement process and exploits the properties of the embedded replacement chain. The joint cumulant generating function of the grade sizes is derived and an asymptotic analysis provides conditions for these to converge in distribution to a multinominal random vector exponentially fast independently of the initial distribution, both for aperiodic and periodic embedded replacement chains. A regenerative approach to the wastage process is outlined and two numerical examples from the literature on manpower planning illustrate the theory.     

On faltung and correlation of functions and their application in physical problems

This paper deals with a number of applications of the correlation and faltung functions, their Fourier transforms and their integrals. It is possible to show that various types of distortions produced by a recording instrument do not affect the value of the integral of the quantity recorded. This should be of great interest to designers of recording instruments. The advantage of using the F.T. in compounding probability distribution functions is pointed out with an illustration giving a short derivation of Kluyver’s famous distribution for the problem of random walk in two dimensions by using this method. Finally, the relation of the correlation function to the Patterson function of a crystal structure is also pointed out.     

Markovian bulk-arrival and bulk-service queues with general state-dependent control

We study a modified Markovian bulk-arrival and bulk-service queue incorporating general state-dependent control. The stopped bulk-arrival and bulk-service queue is first investigated, and the relationship between this stopped queue and the full queueing model is examined and exploited. Using this relationship, the equilibrium behaviour for the full queueing process is studied and the probability generating function of the equilibrium distribution is obtained. Queue length behaviour is also examined, and the Laplace transform of the queue length distribution is presented. The important questions regarding hitting times and busy period distributions are answered in detail, and the Laplace transforms of these distributions are presented. Further properties regarding the busy period distributions including expectation and conditional expectation of busy periods are also explored.

     

The maximum surplus in a finite‐time interval for a discrete‐time risk model with exchangeable,dependent claim occurrences

This paper investigates a discrete‐time risk model that involves exchangeable dependent loss generating claim occurrences and compound binomially distributed aggregate loss amounts. First, a general framework is presented to derive the distribution of a surplus sequence using the model. This framework is then applied to obtain the distribution of any function of a surplus sequence in a finite‐time interval. Specifically, the distribution of the maximum surplus is obtained under nonruin conditions. Based on this distribution, the computation of the minimum surplus distribution is given. Asset and risk management–oriented implications are discussed for the obtained distributions based on numerical evaluations. In addition, comparisons are made involving the corresponding results of the classical discrete‐time compound binomial risk model, for which claim occurrences are independent and identically distributed.     

给定密度随机数的生成的通用方法与球面和球体上均匀随机向量的生成法

本文首先介绍基于垂直概率密度表示的,给定密度函数的随机数生成的通用方法;然后介绍球面及球体上均匀随机向量的生成算法。     

期权定价Black-scholes公式的一个新推导

文章提出Black scholes公式的一个新推导.在风险中性定价的框架下,通过对正态分布的性质及其矩母函数的熟练运用,可以快速容易地导出Black-scholes公式.并且在此推导过程中,公式中的概率值Φ(d1),Φ(d2)的含义得以明确体现.