Statistical inference for spatial and spatio-temporal processes
First, the time series analysis was widely introduced and used in the statistical world. Next, the analysis of spatio-temporal processes has followed, which is taking into account not only when, but also where the phenomenon under observation is taking place. We mainly focus on stationary processes that are assumed to be taking place regularly over both time and space. We examine ways of estimating the parameters involved, without the risk of coming up with a very large bias for our estimators; the bias is the typical problem of estimation for the parameters of stationary processes on Zd, for any d > 2. We particularly study the cases of spatio-temporal ARMA processes and spatial auto-normal formulations on Zd. For both cases and any positive integer d, we propose estimators that are consistent, asymptotically unbiased and normal, if certain conditions are satisfied. We do not only study the spatio-temporal processes that are observed regularly over space, but also those, for which we have recordings on a fixed number of locations anywhere. We might follow the route of a multivariate time series methodology then. Thus, the asymptotic behavior of the estimators proposed might be analyzed as the number of recordings over time only tends to infinity.