TY - THES
A1 - Pezo, Danilo
T1 - Local stationarity for spatial data
N2 - Following the ideas presented in Dahlhaus (2000) and Dahlhaus and Sahm (2000) for time series, we build a Whittle-type approximation of the Gaussian likelihood for locally stationary random fields. To achieve this goal, we extend a Szegö-type formula, for the multidimensional and local stationary case and secondly we derived a set of matrix approximations using elements of the spectral theory of stochastic processes. The minimization of the Whittle likelihood leads to the so-called Whittle estimator \(\widehat{\theta}_{T}\). For the sake of simplicity we assume known mean (without loss of generality zero mean), and hence \(\widehat{\theta}_{T}\) estimates the parameter vector of the covariance matrix \(\Sigma_{\theta}\).
We investigate the asymptotic properties of the Whittle estimate, in particular uniform convergence of the likelihoods, and consistency and Gaussianity of the estimator. A main point is a detailed analysis of the asymptotic bias which is considerably more difficult for random fields than for time series. Furthemore, we prove in case of model misspecification that the minimum of our Whittle likelihood still converges, where the limit is the minimum of the Kullback-Leibler information divergence.
Finally, we evaluate the performance of the Whittle estimator through computational simulations and estimation of conditional autoregressive models, and a real data application.
KW - Räumliche Statistik
KW - Zufälliges Feld
KW - Stochastischer Prozess
KW - Spektralanalyse
KW - Spatial Statistics
KW - Spectral theory
KW - Stochastic Processes
Y1 - 2017
UR - https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5128
UR - https://nbn-resolving.org/urn:nbn:de:hbz:386-kluedo-51287
ER -