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

Consider a stationary Gaussian random field on R d 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.