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