Log-periodogram regression of two-dimensional intrinsically stationary random fields

Abstract

We propose a new semiparametric model for two-dimensional intrinsically stationary random fields and an estimator for the long memory parameter of the model. The model includes a fractional Brownian field, which is isotropic and has been used to model many physical processes in space, as a special case but also allows some anisotropicity. The estimator is based on tapered discrete Fourier transforms and periodograms of observations. Then we apply a log-periodogram regression, which is originally proposed to estimate a long-memory parameter of semiparametric models for time series data. We prove that for our model, the estimator is still consistent and has the limiting normal distribution as the sample size goes to infinity. Furthermore, it is robust to model misspecification. We conduct a computational simulation to compare the performance of it with those of different estimators proposed by other authors and apply our model to an empirical analysis of real data.