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.     

Estimation for nonhomogeneous Poisson processes from aggregated data

A well-known heuristic for estimating the rate function or cumulative rate function of a nonhomogeneous Poisson process assumes that the rate function is piecewise constant on a set of data-independent intervals. We investigate the asymptotic (as the amount of data grows) behavior of this estimator in the case of equal interval widths, and show that it can be transformed into a consistent estimator if the interval lengths shrink at an appropriate rate as the amount of data grows.     

Constructions for planar point processes using concentric circles

Some point processes are obtained by generalising the well-known construction for a two-dimensional Poisson process which locates an event on each of a sequence of concentric circles in a particular way. The constructions considered here have, in general, a random number of events on each circle. Under certain sufficient conditions, the constructed processes are asymptotically Poisson, far from the origin. The obvious regularity in the structure of these processes can be removed at least superficially, by displacing the events independently off the concentric circles.     

On risk processes with the Markov property and with independent increments

We study the influence on the underlying counting process of the Markov property and of the property of independent increments for a risk process.     

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．     

Non-homogeneous space-time fractional Poisson processes

The space-time fractional Poisson process (STFPP), defined by Orsingher and Poilto (2012), is a generalization of the time fractional Poisson process (TFPP) and the space fractional Poisson process (SFPP). We study the fractional generalization of the non-homogeneous Poisson process and call it the non-homogeneous space-time fractional Poisson process (NHSTFPP). We compute their pmf and generating function and investigate the associated differential equation. The limit theorems for the NHSTFPP process are studied. We study the distributional properties, the asymptotic expansion of the correlation function of the non-homogeneous time fractional Poisson process (NHTFPP) and subsequently investigate the long-range dependence (LRD) property of a special NHTFPP. We investigate the limit theorem for the fractional non-homogeneous Poisson process (FNHPP) studied by Leonenko et al. (2014). Finally, we present some simulated sample paths of the NHSTFPP process.     

The largest table in Chinese restaurant processes

In this paper, we investigate the expectation of the size of the largest table in an (α, θ)-Chinese restaurant process by using and developing an idea originated in the work by Shepp, which discusses random permutation. This work was supported by National Natural Science Foundation of China (Grant No. 10671036) and the National Basic Research Program of China (Grant No. 2007CB814904)     

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.     

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.     

Boundary Modified Contact Processes

We introduce a one dimensional contact process for which births to the right of the rightmost particle and to the left of the leftmost particle occur at rate _e (where e is for external). Other births occur at rate _i (where i is for internal). Deaths occur at rate 1. The case _e= _i is the well known basic contact process for which there is a critical value _c>1 such that if the birth rate is larger than _c the process has a positive probability of surviving. Our main motivation here is to understand the relative importance of the external birth rates. We show that if _e1 then the process always dies out while if _e>1 and if _i is large enough then the process may survive. We also show that if _i< _c the process dies out for all _e. To extend this notion to d>1 we introduce a second process that has an epidemiological interpretation. For this process each site can be in one of three states: infected, a susceptible that has never been infected, or a susceptible that has been infected previously. Furthermore, the rates at which the two types of susceptible become infected are different. We obtain some information about the phase diagram about this case as well.     

稀疏过程在破产问题中的应用

本讨论一类人寿保险的风险过程，其中保单到达服从齐次Poisson过程。而描述退保及索赔发生的计数过程分别为这一过程的q-稀疏与p-稀疏．对此模型给出其破产概率的具体上界，并与其它一类风险模型进行比较．     

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.     

Indirect assessment of the bivariate survival function

Estimating the bivariate survival function has been a major goal of many researchers. For that purpose many methods and techniques have been published. However, most of these techniques and methods rely heavily on bivariate failure data. There are situations in which failure time data are difficult to obtain and thus there is a growing need to assess the bivariate survival function for such cases. In this paper we propose two techniques for generating families of bivariate processes for describing several variables that can be used to indirectly assess the bivariate survival function. An estimation procedure is provided and a simulation study is conducted to evaluate the performance of our proposed estimator.     

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.     

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.     

On the Vector Process Obtained by Iterated Integration of the Telegraph Signal

We analyse the vector process (X ₀(t), X ₁(t),...,X _n(t), t > 0) where , and X ₀(t) is the o two-valued telegraph process.In particular, the hyperbolic equations governing the joint distributions of the process are derived and analysed.Special care is given to the case of the process (X ₀(t), X ₁(t), X ₂(t), t > 0) representing a randomly accelerated motion where some explicit results on the probability distribution are derived.     

A class of renewal Interrupted Poisson Processes and applications to queueing systems

Switched Poisson Processes and Interrupted Poisson Processes are often employed to characterize traffic streams in distributed computer and communications systems, especially in investigations of overflow processes in telecommunication networks. With these processes, input streams having inter-segment correlations and high variance as well as state-dependent traffic can properly be modelled. In this paper we first derive an approximation method to describe the Generalized Switched Poisson processes in conjunction with a renewal assumption. As a special case of this class of processes, the class of Interrupted Poisson processes is also included in the investigation. As a result, a generalization of the well-known class of Interrupted Poisson processes is obtained. It is shown that the renewal property is also given for this general class of Interrupted Poisson processes having generally distributed off-phase. To illustrate the accuracy of the presented renewal approximation of Generalized Switched Poisson processes and to show the major properties of the General Interrupted Poisson processes, applications to some basic queueing systems are discussed by means of numerical results.This work was done while the author was with Institute of Communications Switching and Data Technics, University of Stuttgart, Seidenstrasse 36, D-7000 Stuttgart 1, FRG.     

Mixed poisson processes and the probability of ruin

In the Poisson case there is a well known formula that relates the probability of ruin to the distribution function of aggregate claims. It is shown how this formula can be generalized to the mixed Poisson case.     

Risk models based on time series for count random variables

In this paper, we generalize the classical discrete time risk model by introducing a dependence relationship in time between the claim frequencies. The models used are the Poisson autoregressive model and the Poisson moving average model. In particular, the aggregate claim amount and related quantities such as the stop-loss premium, value at risk and tail value at risk are discussed within this framework.