ESTIMATION OF SMOOTHNESS OF A STATIONARY GAUSSIAN RANDOM FIELD

For a stationary Gaussian random field, the decay rate of the spectral density as the frequency becomes large determines the smoothness of the random field. The decay rate of the spectral density is also related to the fractal dimension, which is used to measure the surface smoothness of a random field. We propose an estimator of the decay rate using the periodogram when the observations are on a grid and investigate the asymptotic properties under the fixed domain asymptotic setting. A bias-reduced estimate is proposed based on the theoretical property of the estimator found in this work. A simulation study and a data example are presented.