Composite likelihood inference for space-time covariance models

Modelisation and prediction of environmental phenomena, which typically show dependence in space and time, has been one of the most important challenges over the last years. The classical steps of the spatial modeling approach can be resumed as follows: (1) a model-oriented step, in which a random fields assumption is considered; (2) estimation of the objects defining homogeneity and variability of the random field (i.e the trend and the covariance structure); (3) prediction, which is classically implemented through universal or ordinary kriging procedures. This thesis focuses on the first and second steps. Specifically, it consists of three major parts. First part consists in spatio-temporal covariance modeling. In the geostatistical approach spatial and temporal structure are entangled in the covariance structures and it is not easy to model these two parts simultaneously. The classical kriging predictor, depends crucially on the chosen parametric covariance function. This fact motivates the request for more candidate models of co-variance functions that can be used for space-time data. In particular, the wide variety of practical situations that one may face in the space-time domain motivates the request for flexible models of space-time covariance functions, in order to cover several settings such as non-separability or asymmetry in time. We introduce a new class of stationary space-time covariance model which allows for zonal spatial anisotropies. Second part consists in space and space time covariance models estimation. Maximum likelihood and related techniques are generally considered the best method for estimating the parameters of space-time covariance models. For a spatial Gaussian random field with a given parametric covariance function, exact computation of the likelihood requires calculation of the inverse and determinant of the covariance matrix, and this evaluation is slow when the number of observations is large. The problem increases dramatically in a space-time setting. This fact motivates the search for approximations to the likelihood function that require smaller computational burden and that perform better than classical least square estimation. We introduce a weighted composite likelihood estimation (WCL) for space and space time covariance model estimation. The method induces gains in statistical efficiency with respect to the least squares estimation and from the computational point of view with respect to maximum likelihood estimation. Third part consists in a set of application of WCL. Specifically we apply the method in the estimation of particular spatial covariance functions which allow for negative values and in the estimation of covariance function describing residuals dependence in dynamic life tables. A simulation based test to verify separability of some parametric covariance models in a space time setting is proposed.