New statistical tests for spatial random fields

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A commonly used assumption in the analysis of spatial data is that of (second order) stationarity for the spatial random field used to model the data generating process. This assumption simplifies the mathematical analysis of the model and, as a consequence, substantially facilitates the subsequent development of statistical methodology. Another frequently made assumption to obtain a more powerful statistical analysis is the assumption of isotropy of a stationary random field. However, if stationarity or isotropy are erroneously assumed, then any statistical analysis and conclusion based on these assumptions will be misleading.In this project, we will develop new sophisticated statistical tools to validate the assumption of stationarity (or isotropy) on the basis of a given data set. Our approach is based on the construction of a measure for the deviation of a potentially non-stationary (or non-isotropic) random field from a stationary (or isotropic) random field. The measure vanishes if and if only the assumption of second order stationarity (or isotropy) holds. We construct estimates of this measure as sums of nonlinear functions of local spatial (tapered) periodograms evaluated at a specific set of frequencies and spatial locations and analyze their probabilistic properties such as consistency and weak convergence. In particular, we will study the impact of different sampling schemes and the geometry of the sampling region on the asymptotic behavior (rates of convergence, standardization, asymptotic variances) of the new estimates. Moreover, we will also develop tests for the hypothesis that the deviation from stationarity (or isotropy) is scientifically relevant.