Efficient Estimation of Spectral Functionals for Continuous-Time Stationary Models

Abstract

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 Hajek-Ibragimov-Khas'minskii convolution theorem and Hajek-Le Cam local asymptotic minimax theorem, respectively, and show that the simple "plug-in" statistic ?(I T ), where I T =I T (?) is the periodogram of the underlying stationary Gaussian process X(t) with an unknown spectral density ?(?), ???, is H- and IK-asymptotically efficient estimator for a linear functional ?(?), while for a nonlinear smooth functional ?(?) an H- and IK-asymptotically efficient estimator is the statistic $\Phi(\widehat{\theta}_{T})$ , where $\widehat{\theta}_{T}$ is a suitable sequence of the so-called "undersmoothed" kernel estimators of the unknown spectral density ?(?). Exact asymptotic bounds for minimax mean square risks of estimators of linear functionals are also obtained.