Semiparametric Nonlinear Estimation Of Two-dimensional Long-memory Gaussian Random Fields Perturbed By Noise

In recent years,spatial statistics has been widely used in many fields.The appli-cations of two-dimensional random fields to describe data in image processing,envi-ronment and earth science,spatial econometrics and other fields are ubiquitous.Con-sidering the complexity of two-dimensional spatial data,we combine the classic log-periodogram regression(GPH,stands for Geweke and Porter-Hudak)method and wavelet method with the characteristic of the time-frequency location and multiresolution anal-ysis,to estimate the memory parameters of a two-dimensional Gaussian long memory random field,which is disturbed by an independent,additive and stationary Gaussian short memory noise field.In this thesis,the semiparametric estimations for the memory parameters of one-dimensional long memory time series are extended to the perturbed two-dimensional long memory Gaussian random fields,and then the memory parameters of the random fields are estimated.The thesi mainly contains the following two parts.In the first part,based on the GPH estimation,we consider the semiparametric non-linear estimation of the two memory parameters of the given long memory random field of Type I model with noise disturbance.Two kinds of semiparametric nonlinear log periodogram regression(NLPR)estimators of random fields are proposed,which are based on the GPH estimators.Then,the performance of the three kinds of estimators in finite samples is examined by Monte Carlo simulation,and the impact of the differ-ent values of noise-signal-ratio(nsr),bandwidth and sample size on the estimations is considered.Finally,we compare the performance of the three kinds of estimators.In the second part,based on the wavelet method,we consider the semiparametric nonlinear estimation of the memory parameter of the given long memory random field of Type II model with noise disturbance.The perturbed Gaussian long memory random field and its two-dimensional discrete wavelet transform(DWT)are given at first.And then we obtain the cross-spectral density and variance of the wavelet coefficients of the perturbed random field.After approximating these two,we derive the nonlinear re-gression model about the memory parameter.Then,the nonlinear log-wavelet-variance regression(NLWVR)estimators are proposed,and their consistency with convergence rate,and asymptotic normality are determined.Subsequently,the performance of the NLWVR estimators of the perturbed stationary and intrinsically stationary long memory random fields in finite samples are examined by Monte Carlo simulation,respectively,and the impact of appropriate wavelet selection and different values of noise variance,autoregressive coefficients of short memory component,and sample size on the estima-tion is considered.