In the last years there has been a growing interest in the construction space-time covariance functions. However, effective estimation methods for these models are somehow unexplored. In this paper we propose a composite likelihood approach and a weighted variant for the space-time estimation problem.The proposed method can be a valid compromise between the computational burdens, induced by the use of a maximum likelihood approach, and the loss of efficiency induced by using a weighted least squares procedure. An identification criterion based on the composite likelihood is also introduced. The effectiveness of the proposed procedure is illustrated through an extensive simulation experiment, and by reanalising a data set on Irish wind speeds (Haslett and Raftery, 1989). We also address an important issue, which has been recently explored in the literature, on how to select an appropriate space-time model by accounting for the tradeoff between goodness-of-fit and model complexity.

Spatial analysis methods have seen a rapid rise in popularity due to demand from a wide range of fields. These include, among others, biology, spatial economics, image processing, environmental and earth science, ecology, geography, epidemiology, agronomy, forestry and mineral prospection.In spatial problems, observations come from a spatial process X = {X s , s ∈ S} indexed by a spatial set S, with X s taking values in a state space E. The positions of observation sites s ∈ S are either fixed in advance or random. Classically, S is a 2-dimensional subset, S ⊆ R 2 . However, it could also be 1-dimensional (chromatography, crop trials along rows) or a subset of R 3 (mineral prospection, earth science, 3D imaging). Other fields such as Bayesian statistics and simulation may even require spaces S of dimension d ≥ 3. The study of spatial dynamics adds a temporal dimension, for example (s,t) ∈ R 2 ×R + in the 2-dimensional case. This multitude of situations and applications makes for a very rich subject. To illustrate, let us give a few examples of the three types of spatial data that will be studied in the book.

In the last years there has been a growing interest in proposing methods for estimating covariance functions for geostatistical data. Among these, maximum likelihood estimators have nice features when we deal with a Gaussian model. However maximum likelihood becomes impractical when the number of observations is very large. In this work we review some solutions and we contrast them in terms of loss of statistical efficiency and computational burden. Specifically we focus on three types of weighted composite likelihood functions based on pairs and we compare them with the method of covariance tapering. Asymptotics properties of the three estimation methods are derived. We illustrate the effectiveness of the methods through theoretical examples, simulation experiments and by analysing a data set on yearly total precipitation anomalies at weather stations in the United States.

This paper presents a hierarchical approach to modelling extremes of a stationary time series. The procedure comprises two stages. In the first stage, exceedances over a high threshold are modelled through a generalized Pareto distribution, which is represented as a mixture of an exponential variable with a Gamma distributed rate parameter. In the second stage, a latent Gamma process is embedded inside the exponential distribution in order to induce temporal dependence among exceedances. Unlike other hierarchical extreme-value models, this version has marginal distributions that belong to the generalized Pareto family, so that the classical extremevalue paradigm is respected. In addition, analytical developments show that different choices of the underlying Gamma process can lead to different degrees of temporal dependence of extremes, including asymptotic independence. The model is tested through a simulation study in a Markov chain setting and used for the analysis of two datasets, one environmental and one financial. In both cases, a good flexibility in capturing different types of tail behaviour is obtained.

Max-stable processes play a fundamental role in modeling the spatial dependence of extremes because they appear as a natural extension of multivariate extreme value distributions. In practice, a well-known restrictive assumption when using max-stable processes comes from the fact that the observed extremal dependence is assumed to be related to a particular max-stable dependence structure. As a consequence, the latter is imposed to all events which are more extreme than those that have been observed. Such an assumption is inappropriate in the case of asymptotic independence. Following recent advances in the literature, we exploit a max-mixture model to suggest a general spatial model which ensures extremal dependence at small distances, possible independence at large distances and asymptotic independence at intermediate distances.Parametric inference is carried out using a pairwise composite likelihood approach. Finally we apply our modeling framework to analyze daily precipitations over the East of Australia, using block maxima over the observation period and exceedances over a large threshold.2