Efficient estimation of spectral functionals for Gaussian stationary models
Mamikon Smbatovich Ginovyan · Communications on Stochastic Analysis · 2011
The paper considers a problem of construction of asymptotically efficient estimators for spectral functionals, and bounding the minimax mean square risks.We consider the efficiency concepts of estimators, based on the variants of Hájek-Ibragimov-Khas'minskii convolution theorem (H-efficiency) and Hájek-Le Cam local asymptotic minimax theorem (IK-efficiency), and show that the simple "plug-in" statistic Φ(I T ), where I T = I T (λ) is the periodogram of the underlying stationary Gaussian process X(t) with an unknown spectral density θ(λ), is H-and IK-asymptotically efficient estimator for a linear functional Φ(θ), while for a nonlinear smooth functional Φ(θ) an H-and IK-asymptotically efficient estimator is the statistic Φ( θ T ), where θ T is a sequence of "undersmoothed" kernel estimators of the unknown spectral density θ(λ).Exact asymptotic bounds for minimax mean square risks of estimators of linear functionals are also obtained.