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STATISTICAL ANALYSIS OF LONG-RANGE DEPENDENT RANDOM PROCESSES AND FIELDS

The thesis deals with estimation of long-memory parameters of random processes and fields. Long-memory behaviour of data has been reported in various applications, for example in finance, internet traffic and hydrology. It is known that estimators of parameters of long-memory and especially seasonal long-memory processes have complex non-Gaussian asymptotic distributions (see [5]). In the first part, the consistency and asymptotic normality for the so-called Ibragimov minimum contrast estimators based on tapered data are investigated for various models of Gaussian random processes and fields. Periodogram-based estimators of parameters are used. Tapering is employed to reduce the bias of the periodogram in the spatial data. It is known that tapering data reduces leakage effects, especially when spectral densities contain high peaks. The use of tapers can lead to bias reduction, which is a key issue in spatial statistics, by helping to remove the socalled ‘edge effects’. The thesis develops a new methodology to verify assumptions for consistency and asymptotic normality of long-memory parameter estimators. The cases of Gaussian fractional autoregressive processes and Gegenbauer random fields are considered. This methodology is applicable for shortand long-range dependent cases. We review some results from the literature on the Whittle estimators based on tapered data, with a particular focus on the conditions on tapers which help to control the bias of estimators. Then we study the case of the Whittle estimators with tapered data for weakly and strongly dependent random fields. It is proved that minimum contrast estimators of the parameters are consistent estimators. Various particular cases of spectral densities and tapers are considered.