Efficient estimation of spectral functionals for Gaussian stationary models

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(IT ), where IT = IT (λ) is the periodogram of the underlying stationary Gaussian process X(t) with an unknown spectral density θ(λ), is Hand IK-asymptotically efficient estimator for a linear functional Φ(θ), while for a nonlinear smooth functional Φ(θ) an Hand IK-asymptotically efficient estimator is the statistic Φ(θ̂T ), where θ̂T 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.