Locally Rectified Whittle Likelihood Method Based On Spatial Grid Data

The likelihood method based on large-scale spatial data on multi-dimensional grids often has the problem of computational difficulty because it involves the inversion of covariance ma-trix.The Whittle likelihood function,which is set up from the periodogram of the observations,provides a good approximation to the true likelihood.Using the fast Fourier transform,the computation in Whittle likelihood method is very efficient,this approach may produce biased estimators.Combining the ideas of kernel-function-weighting and de-biasing.Therefore,it is of great importance to improve the performance of Whittle likelihood by de-biasing.Combining the ideas of kernel-function-weighting and de-biasing,a local de-biased Whit-tle likelihood method,which is built up based on the expectation of periodogram,is proposed.The main works are as follows.First,a local de-biased Whittle likelihood function is defined under the assumption of a certain parametric spectral density model.The model parameter is estimated by minimizing the local de-biased Whittle likelihood function,and the spectral den-sity estimator can be obtained by substituting the parameter estimator into the spectral density function.For the parameter estimation and spectral density estimation,we prove their asymp-totic normality under some regularity conditions.Second,the hypothesis testing problem of spectral density is considered.Based on the spectral density estimation and the properties of periodogram,a local de-biased Whittle likelihood ratio test statistic is constructed,and the sam-pling distribution of the statistic is given for the goodness-of-fit test of spectral density.Third,in order to investigate the performance of the proposed approach under the small sample,the simulation study is designed.The simulation results show that the proposed de-biased method is effective.Compared with the local Whittle likelihood method,the proposed method has better performance in spectral density parameter inference.Last,the wheat yield data of each block on the isometric 20 x 25 are employed for empirical analysis.The analysis results show that the model parameter estimate obtained by the proposed approach is closer to the one obtained by the maximum likelihood method.This paper is concerned with the de-biasing in the local Whittle likelihood method.The purpose is fulfilled by defining a novel local Whittle likelihood function,in which,the spectral density in the Whittle likelihood is replaced by the expectation of periodogram.By the scheme,both the bias and the mean square error of the estimator is reduced.As a result,the local Whittle likelihood is improved.