On Estimation of Poisson Intensity Functions

Under the presence of only one realization, we consider a computationally simple algorithm for estimating the intensity function of a Poisson process with exponential quadratic and cyclic of fixed frequency trends. We argue that the algorithm can successfully be used to estimate any Poisson intensity function provided that it has a parametric form.     

泊松点过程的若干大偏差估计

本文给出泊松点过程下列三种极限行为的大偏差估计：（１）高密度情形；（２）低密度情形和（３）标度变换下极限情形。     

A New Approach to Poisson Approximation of Simple Point Processes Using Compensators

Consider a simple point process N on the line, and let be its compensator. We use a result of Kallenberg (1990, Probab. Theory Relat. Fields 86, 167–202) to give a new approach to estimate the total variation distance between the distributions of N and that of a Poisson process when has small jump sizes.     

On Exit Times of a Multivariate Random Walk and Its Embedding in a Quasi Poisson Process

Abstract

We introduce and analyze a delayed renewal process  = {τ₀,τ₁,…} marked by a multivariate random walk (,) and its behavior about fixed levels to be crossed by one of the components of (,). We derive the joint distribution of first passage time τ_ρ, pre-exit time τ_ρ?1 (i.e., the instant one phase prior to the first passage time), and the respective values of (,) at τ_ρ and τ_ρ?1 in a closed form. The results obtained are then applied to a multivariate quasi Poisson process Π, forming a random walk ((Π),) embedded in Π over . Processes like these can model various phenomena including stock market and option trading.

One of the central issues in the investigation of ((Π),) is to obtain the information about Π at any moment of time in random vicinities of τ_ρ and τ_ρ?1 previously available only upon . The results offer, again, closed form functionals. Numerous examples throughout the paper illustrate introduced constructions and connect the results with real-world applications, most prominently the stock market.     

Robustness for Inhomogeneous Poisson Point Processes

We consider robustness for estimation of parametric inhomogeneous Poisson point processes. We propose an influence functional to measure the effect of contamination on estimates. We also propose an M-estimator as an alternative to maximum likelihood estimator, show its consistency and asymptotic normality.     

On Polygons Enclosing Point Sets II

Let R and B be disjoint point sets such that R ∪ B is in general position. We say that B encloses by R if there is a simple polygon P with vertex set B such that all the elements in R belong to the interior of P. In this paper we prove that if the vertices of the convex hull of R ∪ B belong to B, and |R| ≤ |Conv(B)| − 1 then B encloses R. The bound is tight. This improves on results of a previous paper in which it was proved that if |R| ≤ 56 |Conv(B)| then B encloses R. To obtain our result we prove the next result which is interesting on its own right: Let P be a convex polygon with n vertices p ₁ , ... , p _n and S a set of m points contained in the interior of P, m ≤ n − 1. Then there is a convex decomposition {P ₁ , ... , P _n } of P such that all points from S lie on the boundaries of P ₁ , ... , P _n , and each P _i contains a whole edge of P on its boundary. F. Hurtado partially supported by projects MEC MTM2006-01267 and DURSI 2005SGR00692. C. Merino supported by CONACYT of Mexico, Proyecto 43098. J. Urrutia supported by CONACYT of Mexico, Proyecto SEP-2004-Co1-45876, and MCYT BFM2003-04062. I. Ventura partially supported by Project MCYT BFM2003-04062.     

On the central limit theorem for the stationary Poisson process of compact sets

Stochastic geometry models based on a stationary Poisson point process of compact subsets of the Euclidean space are examined. Random measures on ?^d, derived from these processes using Hausdorff and projection measures are studied. The central limit theorem is formulated in a way which enables comparison of the various estimators of the intensity of the produced random measures. Approximate confidence intervals for the intensity are constructed. Their use is demonstrated in an example of length intensity estimation for the segment processes. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

极值过程的跳时点过程的精致渐近性

以统一的形式,得到了极值过程的跳时点过程的精致渐近性的一般结果,揭示了精致渐近性研究中拟权函数,边界函数,收敛速度以及极限状态之间的有机关系,从而可以给出许多新的具体结果。     

On Rescaled Poisson Processes and the Brownian Bridge

The process obtained by rescaling a homogeneous Poisson process by the maximum likelihood estimate of its intensity is shown to have surprisingly strong self-correcting behavior. Formulas for the conditional intensity and moments of the rescaled Poisson process are derived, and its behavior is demonstrated using simulations. Relationships to the Brownian bridge are explored, and implications for point process residual analysis are discussed.     

防洪风险分析中的一个二元标值犘狅犻狊狊狅狀点过程模型

该文将洪水的大小和持续时间作为防洪设施的工程风险中不可忽略的因素,提出了以洪水的大小和持续时间为标值的二元标值Ｐｏｉｓｓｏｎ点过程模型,给出了防洪综合风险率的计算公式,并进行了实例计算．     

某些一致分布的点集序列

本文借助于“倒根函数”和矩阵构造定义了［０,１）Ｓ（Ｓ≥１）中的一些有限点集,给出了它们的偏差的上界估计,从而证明了由它们组成的点集序列是一致分布的．     

On a Class of Uniformly Distributed Sequences of Point Sets

In this note we construct certain sequences of finite point sets in [0, 1)^s(s ≥ 1) and give the upper bounds of their discrepancy. Furthermore we prove that these sequences are uniformly distrbuted in [0, 1)^s.     

Two Ergodic Sample-Path Properties of the Poisson Process

For increasing sequences of real numbers we consider two types of asymptotic behavior that remind of the defining property of a (homogeneous) Poisson process according to which the numbers of points in disjoint intervals are independent and follow Poisson distributions with specified parameters. We prove that almost all paths of a Poisson process show this asymptotic behavior, and characterize the Poisson process by these properties. Further we discuss the connection to equidistribution notions.     

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.     

保费收入为Poisson过程的更新风险模型

对于保费收入为Poisson过程的更新风险模型,利用马氏链的理论,借助转移概率,得出了破产概率和破产赤字的展式及其所满足的积分方程.     

保险费收取次数为泊松过程下的广义复合泊松风险模型

经典的破产模型是假定保险公司按单位时间常数速率收取保险费，盈余过程{R（t），t≥0中的S（f）=∑i=1^N（t）Y,为一复合泊松过程，本文将保费到达过程推广为一个Poisson过程，同时将S（t）推广为一个广义复合Poisson过程．针对此模型给出了盈余过程的一些性质，得到关于破产概率的一个定理．     

Continuity and Boundedness of Infinitely Divisible Processes: A Poisson Point Process Approach

Sufficient conditions for boundedness and continuity are obtained for stochastically continuous infinitely divisible processes, without Gaussian component, {Y(t),t T}, where T is a compact metric space or pseudo-metric space. Such processes have a version given by Y(t)=X(t)+b(t),tT where b is a deterministic drift function and

A Two-Sided First-Exit Problem for a Compound Poisson Process with a Random Upper Boundary

We consider the first-exit time of a compound Poisson process from a region that is bounded from below by an increasing straight line, while its upper boundary has positive jumps of i.i.d. sizes at Poisson times and increases linearly between jumps. An integral equation for the corresponding Laplace-Stieltjes transforms is derived and solved. The case of exponential jumps is treated separately. The problem has applications in queueing and risk theory.AMS 2000 Mathematics Subject Classification: Primary 60G40; Secondary 62K25, 6OJ75     

Classification of Mixtures of Spatial Point Processes via Partial Bayes Factors

Motivated by the problem of minefield detection, we investigate the problem of classifying mixtures of spatial point processes. In particular we are interested in testing the hypothesis that a given dataset was generated by a Poisson process versus a mixture of a Poisson process and a hard-core Strauss process. We propose testing this hypothesis by comparing the evidence for each model by using partial Bayes factors. We use the term partial Bayes factor to describe a Bayes factor, a ratio of integrated likelihoods, based on only part of the available information, namely that information contained in a small number of functionals of the data. We applied our method to both real and simulated data, and considering the difficulty of classifying these point patterns by eye, our approach overall produced good results.