Local Whittle estimator for anisotropic random fields

AMS 1991 subject classifications: primary 62G07 secondary 62M10 Keywords: Spatial long memory Local Whittle method a b s t r a c t A local Whittle estimator is developed to simultaneously estimate the long memory parameters for stationary anisotropic scalar random fields. It is shown that these estimators are consistent and asymptotically normal, under some weak technical conditions. A brief simulation study illustrates a practical application of the estimator. Stationary scalar random fields with spatial long memory are useful in many diverse areas of applications (see, e.g., [2,8,21] and references therein). The Hurst parameter codes the extent of long memory, i.e., the power-law decay of autocorrelation as a function of separation distance in space. In many applications, it is unreasonable to employ an isotropic model, and hence there is a different Hurst index in each coordinate direction. In studies of ground water flow and contaminant transport, essential physical properties such as hydraulic conductivity are commonly modelled as scalar random fields with long memory. Estimates of the Hurst (long memory) index typically yield a larger value in the direction of flow, and a smaller value in the direction transverse to the flow. In this paper, we develop a robust method to simultaneously estimate the Hurst index in each scaling direction. Our local Whittle estimator is based on spectral methods, essentially the idea that the power spectrum grows as a power law near zero if the autocorrelation decays as a power law near infinity. If the autocorrelations decay at a different power law rate in each spatial coordinate direction, then the spectral density grows as a different power law in each coordinate of the frequency. The local Whittle method assumes only the power-law asymptotics of the spectral density at the origin, making it extremely robust. The usual Whittle estimator estimates the Hurst index using the entire spectral density, and consequently the bias and standard deviation of the full Whittle estimator are comparable. One advantage of the local Whittle method is that the bias is always negligible with respect to the standard deviation, see Guyon [15]. Most commonly used random field models with long memory are isotropic [1,27]. The prototypical example is the fractional Brownian random field with moving average representation B(x) = R d x − y H−d/2 − y H−d/2 W (dy) (1.1) where 0 < H < 1 and W (dy) is an independently scattered Gaussian random measure on R …