Tail estimation of the spectral density for a stationary Gaussian random field

Consider a stationary Gaussian random field on Rd with spectral density f(λ) that satisfies f(λ)∼c|λ|−θ as |λ|→∞. The parameters c and θ control the tail behavior of the spectral density. c is related to a microergodic parameter and θ is related to a fractal index. For data observed on a grid, we propose estimators of c and θ by minimizing an objective function, which can be viewed as a weighted local Whittle likelihood, study their properties under the fixed-domain asymptotics and provide simulation results.