Efficient Estimation of Spectral Functionals for Continuous-Time Stationary Models

The paper considers a problem of construction of asymptotically efficient estimators for functionals defined on a class of spectral densities, and bounding the minimax mean square risks. We define the concepts of H- and IK-efficiency of estimators, based on the variants of Hájek-Ibragimov-Khas’minskii convolution theorem and Hájek-Le Cam local asymptotic minimax theorem, respectively, 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), λ∈ℝ, 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 F(^(q)]_T)\Phi(\widehat{\theta}_{T}), where ^(q)]_T\widehat{\theta}_{T} is a suitable sequence of the so-called “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.