Limit theorems for functionals on the facets of stationary random tessellations

We observe stationary random tessellations $X=\{\Xi_n\}_{n\ge1}$ in $\mathbb{R}^d$ through a convex sampling window $W$ that expands unboundedly and we determine the total $(k-1)$-volume of those $(k-1)$-dimensional manifold processes which are induced on the $k$-facets of $X$ ($1\le k\le d-1$) by their intersections with the $(d-1)$-facets of independent and identically distributed motion-invariant tessellations $X_n$ generated within each cell $\Xi_n$ of $X$. The cases of $X$ being either a Poisson hyperplane tessellation or a random tessellation with weak dependences are treated separately. In both cases, however, we obtain that all of the total volumes measured in $W$ are approximately normally distributed when $W$ is sufficiently large. Structural formulae for mean values and asymptotic variances are derived and explicit numerical values are given for planar Poisson--Voronoi tessellations (PVTs) and Poisson line tessellations (PLTs).Comment: Published at http://dx.doi.org/10.3150/07-BEJ6131 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

A Sliding Blocks Estimator for the Extremal Index

In extreme value statistics for stationary sequences, blocks estimators are usually constructed by using disjoint blocks because exceedances over high thresholds of different blocks can be assumed asymptotically independent. In this paper we focus on the estimation of the extremal index which measures the degree of clustering of extremes. We consider disjoint and sliding blocks estimators and compare their asymptotic properties. In particular we show that the sliding blocks estimator is more efficient than the disjoint version and has a smaller asymptotic bias. Moreover we propose a method to reduce its bias when considering sufficiently large block sizes.Comment: Submitted to the Electronic Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of Mathematical Statistics (http://www.imstat.org

Methods for estimating the upcrossings index: improvements and comparison

The upcrossings index 0≤η≤1, as a measure of the degree of local dependence in the upcrossings of a high level by a stationary process, plays, together with the extremal index θ, an important role in extreme events modelling. For stationary processes, verifying a long range dependence condition, upcrossings of high thresholds in different blocks can be assumed asymptotically independent and therefore blocks estimators for the upcrossings index can be easily constructed using disjoint blocks. In this paper we focus on the estimation of the upcrossings index via the blocks method and properties such as consistency and asymptotic normality are studied. Besides this new estimation approach for this parameter, we also enlarge its family of runs estimators and improve estimation within this class by providing an empirical way of checking local dependence conditions that control the clustering of upcrossings. We compare the performance of a range of different estimators for η and illustrate the methods using simulated data and financial data.info:eu-repo/semantics/publishedVersio

Nonparametric inference for first-order characteristics of spatial and spatio-temporal point processes

A point process is a stochastic process that generates a random collection of events in some metric space. A spatial point process generates events on a planar bounded region. If in addition to the spatial location of the events we know the time of occurrence we have a spatio-temporal point pattern. Spatial and spatio-temporal point patterns arise in a wide variety of scientific contexts, including ecology, seismology, epidemiology, cosmology and geography. In particular, the spatial location and time of occurrence of wildfires, the main threat for forests around the world, can be seen as spatio-temporal point patterns. The first-order intensity function characterizes the structure of events and is needed to estimate the second-order characteristics, which describe interaction between events. For these reasons modeling the first-order intensity function is a main issue in the analysis of both spatial and spatio-temporal point processes, and kernel estimators with scalar bandwidth have been widely used to this purpose. In the spatial framework, this work focuses on the consistent kernel intensity estimator with full matrix bandwidth. We develop an effective smooth bootstrap procedure which allows to estimate consistently the MISE of the consistent kernel intensity estimator, and we suggest a procedure to select the optimal bandwidth matrix. First-order separability is commonly assumed to estimate the spatio-temporal intensity function without being formally tested. This work proposes nonparametric separability tests for the intensity function of spatio-temporal point processes with discrete temporal component, which can be seen as multitype spatial point processes, and with continuous temporal component. The different techniques developed in this work have been applied to the analysis of wildfires registered in Galicia along the period 1999-2008

Spatial marked point processes: Models and inferences

A spatial marked point process describes the locations of randomly distributed events in a region, with a mark attached to each observed point. Nowadays, the availability of spatiotemporal data is increasing and many spatiotemporal models are studied with applications in a wide range of disciplines. Spatial marked point processes are then extended to spatiotemporal marked point processes if time component is taken into account. In general, the marks can be quantitative or categorical variables. Independence between points and marks is a convenient assumption, but may not be true in practice. Tests for independence between points and marks are proposed previously, though only a few models have been developed to describe dependence between points and marks. In this dissertation, I focus on quantitative marks and the objective is to provide flexible models for both spatial and spatiotemporal marked point processes when points and marks are dependent.^ Three approaches to describe dependence between points and marks are studied in this dissertation, while the first two approaches are for spatial marked point processes and the last is for spatiotemporal marked point processes. First, we derive a covariance function of additive models for marked point processes. This covariance function carries information of dependence between points and marks, which can be used in kriging to make predictions of marks at unknown locations. We expect to obtain better prediction results by using this covariance function when the points and marks are dependent.^ The second approach is to consider intensity-dependent models. We study both univariate and bivariate intensity marked Log Gaussian Cox processes and apply an empirical Bayesian estimation procedure with implementation of Markov Chain Monte Carlo methodology for statistical inference. We allow dependence between marks after conditioning on the intensity which is more flexible than conditional independence assumption. The influence of adding cross covariance in modeling bivariate marks is also explored. The first two approaches are applied to model the dependence between points and marks of a white oak data.^ The last approach is to consider the partially stationary spatiotemporal marked point process, where the distribution of the spatiotemporal marked process is invariant under parallel shift of time, but may not be invariant under parallel shift of points or marks. It can be classified as a location-dependent model. To determine the potential usefulness of this approach, we illustrate through two typical examples in natural hazards: a forest wild fire study and an earthquake study. The results show that the distribution of marks and points is significantly different at local scale. It is expected that the proposed approach will have wide applications in the study of natural hazards