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

Consider a stationary Gaussian random field on RdRd with spectral density f(λ) that satisfies f(λ)∼c|λ|−θ as |λ|→∞. The parameters cc and θθ control the tail behavior of the spectral density. cc is related to a microergodic parameter and θθ is related to a fractal index. For data observed on a grid, we propose estimators of cc and θθ 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.