### 50 years sets with positive reach -- a survey.

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Prediction of outstanding liabilities is an important problem in non-life insurance. In the framework of the Solvency II Project, the best estimate must be derived by well defined probabilistic models properly calibrated on the relevant claims experience. A general model along these lines was proposed earlier by Norberg (1993, 1999), who suggested modelling claim arrivals and payment streams as a marked point process. In this paper we specify that claims occur in [0,1] according to a Poisson point...

The problem of estimating the intensity of a non-stationary Poisson point process arises in many applications. Besides non parametric solutions, e. g. kernel estimators, parametric methods based on maximum likelihood estimation are of interest. In the present paper we have developed an approach in which the parametric function is represented by two-dimensional beta-splines.

We give a universal discrimination procedure for determining if a sample point drawn from an ergodic and stationary simple point process on the line with finite intensity comes from a homogeneous Poisson process with an unknown parameter. Presented with the sample on the interval $[0,t]$ the discrimination procedure ${g}_{t}$, which is a function of the finite subsets of $[0,t]$, will almost surely eventually stabilize on either POISSON or NOTPOISSON with the first alternative occurring if and only if the process is...

We propose in this work an original estimator of the conditional intensity of a marker-dependent counting process, that is, a counting process with covariates. We use model selection methods and provide a nonasymptotic bound for the risk of our estimator on a compact set. We show that our estimator reaches automatically a convergence rate over a functional class with a given (unknown) anisotropic regularity. Then, we prove a lower bound which establishes that this rate is optimal. Lastly, we provide...

The contribution deals with an application of the nonparametric version of Cox regression model to the analysis and modeling of the failure rate of technical devices. The objective is to recall the method of statistical analysis of such a model, to adapt it to the real–case study, and in such a way to demonstrate the flexibility of the Cox model. The goodness-of-fit of the model is tested, too, with the aid of the graphical test procedure based on generalized residuals.

We consider a flexible class of space-time point process models—inhomogeneous shot-noise Cox point processes. They are suitable for modelling clustering phenomena, e.g. in epidemiology, seismology, etc. The particular structure of the model enables the use of projections to the spatial and temporal domain. They are used to formulate a step-wise estimation method to estimate different parts of the model separately. In the first step, the Poisson likelihood approach is used to estimate the inhomogeneity...

A new point process is proposed which can be viewed either as a Boolean cluster model with two cluster modes or as a $p$-thinned Neyman-Scott cluster process with the retention of the original parent point. Voronoi tessellation generated by such a point process has extremely high coefficients of variation of cell volumes as well as of profile areas and lengths in the planar and line induced tessellations. An approximate numerical model of tessellation characteristics is developed for the case of small...

This article provides entropic inequalities for binomial-Poisson distributions, derived from the two point space. They appear as local inequalities of the M/M/∞ queue. They describe in particular the exponential dissipation of Φ-entropies along this process. This simple queueing process appears as a model of “constant curvature”, and plays for the simple Poisson process the role played by the Ornstein-Uhlenbeck process for Brownian Motion. Some of the inequalities are recovered by semi-group ...

Boolean cluster point processes with various cluster distributions are examined by means of their spherical contact distribution function. Special attention is paid to clusters with strong independence properties (Poisson clusters) and regular clusters.

A method known as renormalization is proposed for constructing some more physically realistic random potentials in a Poisson cloud. The Brownian motion in the renormalized random potential and related parabolic Anderson models are modeled. With the renormalization, for example, the models consistent to Newton’s law of universal attraction can be rigorously constructed.