Model testing for spatially correlated data

The distribution of a variable observed over a domain depends on the underlying process and also on the geographical locations at which the variable has been measured. In this paper, we fit a model to the distribution supposing that the observations are generated by a stationary strong-mixing random field. Indeed, after estimating the density of the considered variable, we construct a test statistic in order to verify the goodness of fit of the observed spatial data. The proposed class of tests is a generalization of the classical chi-square-test and of the Neyman smooth test. In the framework of increasing domain asymptotics, we analyse the large sample behaviour of the test. The limiting distribution is a linear combination of χ2 r.v.s where the coefficients are the eigenvalues of a matrix Σ essentially related to the spectral density of the random field. Finally some indications about the implementation are provided.