We formulate and evaluate weighted and ordinary least squares procedures for estimating the parametric rate function of a nonhomogeneous Poisson process. Special emphasis is given to processes having an exponential rate function, where the exponent may include a polynomial component or some trigonometric components or both. Theoretical and experimental evidence is provided to explain some surpr...

Fractional generalizations of the Poisson process and branching Furry process are considered. The link between characteristics of the processes, fractional differential equations and Lèvy stable densities are discussed and used for construction of the Monte Carlo algorithm for simulation of random waiting times in fractional processes. Numerical calculations are performed and limit distribution...

As time passes, change occurs. With this change comes the need for surveillance. One may be a technician on an assembly line and in need of a surveillance technique to monitor the number of defective components produced. On the other hand, one may be an administrator of a hospital in need of surveillance measures to monitor the number of patient falls in the hospital or to monitor surgical outc...

We present the explicit solution of the Bayesian problem of sequential testing of two simple hypotheses about the intensity of an observed Poisson process. The method of proof consists of reducing the initial problem to a free-boundary differential-difference Stephan problem, and solving the latter by use of the principles of smooth and continuous fit. A rigorous proof of the optimality of the ...

The definition of vectors of dependent random probability measures is a topic of interest in applications to Bayesian statistics. They, indeed, represent dependent nonparametric prior distributions that are useful for modelling observables for which specific covariate values are known. In this paper we propose a vector of two-parameter Poisson-Dirichlet processes. It is well-known that each com...

Poisson cluster processes are one of the most important classes of point process models (see Daley and Vere-Jones[7] and Møller; Waagepetersen[24]). They are natural models for the location of objects in the space, and are widely used in point process studies whether theoretical or applied. Very popular and versatile Poisson cluster processes are the so-called self-exciting or Hawkes processes ...

We announce a deterministic analog of Bartlett’s displacement theorem. The result is that a Poisson property is stable with respect to deterministic Hamiltonian displacements. While the random point configurations move according to an n-body evolution, the mean measure P satisfies a nonlinear Vlasov type equation Ṗ + y · ∇xP − ∇y · E(P ) = 0. Combined with Bartlett’s theorem, the result general...

Poisson processes on xed intervals are imbedded in simulations where the goal is to estimate the expectation of some function of the process and of other random variables. Quasi-Monte Carlo is used, in an eecent way, to generate stage-wise (in tree-like fashion) certain non-consecutive arrival epochs with well-spaced indices while the others are then generated as needed by standard Monte Carlo ...

It is our intention to provide via fractional calculus a generalization of the pure and compound Poisson processes, which are known to play a fundamental role in renewal theory, without and with reward, respectively. We first recall the basic renewal theory including its fundamental concepts like waiting time between events, the survival probability, the counting function. If the waiting time i...

A branching process is a mathematical description of the growth of a population for which the individual produces offsprings according to stochastic laws. A typical problem would be of the following form. Considering a population of individuals developing from a single progenitor, the initial individual. The initial individual produces a random number of offsprings, each of them in turn produce...