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Local stationarity for spatial data

  • 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.

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Author:Danilo Pezo
URN (permanent link):urn:nbn:de:hbz:386-kluedo-51287
Advisor:Jürgen Franke, Rainer Dahlhaus
Document Type:Doctoral Thesis
Language of publication:English
Publication Date:2018/01/16
Year of Publication:2017
Publishing Institute:Technische Universität Kaiserslautern
Granting Institute:Technische Universität Kaiserslautern
Acceptance Date of the Thesis:2017/12/20
Date of the Publication (Server):2018/01/17
Tag:Spatial Statistics; Spectral theory; Stochastic Processes
GND-Keyword:Räumliche Statistik; Spektralanalyse <Stochastik>; Stochastischer Prozess; Zufälliges Feld
Number of page:XV, 79
Faculties / Organisational entities:Fachbereich Mathematik
DDC-Cassification:5 Naturwissenschaften und Mathematik / 510 Mathematik
MSC-Classification (mathematics):62-XX STATISTICS / 62Hxx Multivariate analysis [See also 60Exx] / 62H11 Directional data; spatial statistics
Licence (German):Creative Commons 4.0 - Namensnennung, nicht kommerziell (CC BY-NC 4.0)