Summary characteristics play an important role in the analysis of spatial point processes. We discuss various approaches to estimating summary characteristics from replicated observations of a stationary point process. The estimators are compared with respect to their integrated squared error. Simulations for three basic types of point processes help to indicate the best way of pooling the subwindow estimators. The most appropriate way depends on the particular summary characteristic, edge-correction...

A histogram sieve estimator of the drift function in Ito processes and some semimartingales is constructed. It is proved that the estimator is pointwise and L¹ consistent and its finite-dimensional distributions are asymptotically normal. Our approach extends the results of Leśkow and Różański (1989a).

In the article, we propose a new estimator of the hazard rate function in the framework of the multiplicative point process intensity model. The technique combines the reflection method and the method of transformation. The new method eliminates the boundary effect for suitably selected transformations reducing the bias at the boundary and keeping the asymptotics of the variance. The transformation depends on a pre-estimate of the logarithmic derivative of the hazard function at the boundary.

A complete and sufficient statistic is found for stationary marked Poisson processes with a parametric distribution of marks. Then this statistic is used to derive the uniformly best unbiased estimator for the length density of a Poisson or Cox segment process with a parametric primary grain distribution. It is the number of segments with reference point within the sampling window divided by the window volume and multiplied by the uniformly best unbiased estimator of the mean segment length.

We investigate estimators of the asymptotic variance ${\sigma }^2$ of a $d$–dimensional stationary point process $\Psi$ which can be observed in convex and compact sampling window $W_n=nW$. Asymptotic variance of $\Psi$ is defined by the asymptotic relation $Var\left(\Psi \left(W_n\right)\right)\sim {\sigma }^2\left|W_n\right|$ (as $n\to \infty$) and its existence is guaranteed whenever the corresponding reduced covariance measure ${\gamma }_{\mathrm{red}}^{\left(2\right)}(·)$ has finite total variation. The three estimators discussed in the paper are the kernel estimator, the estimator based on the second order intesity of the point process and the...

Classical extreme value methods were derived when the underlying process is assumed to be a sequence of independent random variables. However when observations are taken along the time and/or the space the independence is an unrealistic assumption. A parameter that arises in this situation, characterizing the degree of local dependence in the extremes of a stationary series, is the extremal index, θ. In several areas such as hydrology, telecommunications, finance and environment, for example, the...