In this talk, we consider function-indexed normalized weighted integrated periodograms for equidistantly sampled multivariate continuous-time state space models which are multivariate continuous-time ARMA processes. Thereby, the sampling distance is fixed and the driving Lévy process has at least a finite fourth moment. Under different assumptions on the function space and the moments of the driving Lévy process we derive a central limit theorem for the function-indexed normalized weighted integrated periodogram. Either the assumption on the function space or the assumption on the existence of moments of the Lévy process is weaker. The results can be used to derive the asymptotic behavior of the Whittle estimator and to construct goodness-of-fit test statistics as the Grenander-Rosenblatt statistic and the Cramér-von Mises statistic.[-]

In this talk we introduce a class of statistics for spatial data that is observed on an irregular set of locations. Our aim is to obtain a unified framework for inference and the statistics we consider include both parametric and nonparametric estimators of the spatial covariance function, Whittle likelihood estimation, goodness of fit tests and a test for second order spatial stationarity. To ensure that the statistics are computationally feasible they are defined within the Fourier domain, and in most cases can be expressed as a quadratic form of a discrete Fourier-type transform of the spatial data. Evaluation of such statistic is computationally tractable, requiring $O(nb)$ operations, where $b$ are the number Fourier frequencies used in the definition of the statistic (which varies according to the application) and $n$ is the sample size. The asymptotic sampling properties of the statistics are derived using mixed spatial asymptotics, where the number of locations grows at a faster rate than the size of the spatial domain and under the assumption that the spatial random field is stationary and the irregular design of the locations are independent, identically distributed random variables. We show that there are quite intriguing differences in the behaviour of the statistic when the spatial process is Gaussian and non-Gaussian. In particular, the choice of the number of frequencies $b$ in the construction of the statistic depends on whether the spatial process is Gaussian or not. If time permits we describe how the results can also be used in variance estimation. And if we still have time some simulations and real data will be presented.[-]