Asymptotically Efficient Nonparametric Estimation of Nonlinear Spectral Functionals

Abstract

The paper considers a problem of construction of asymptotically efficient estimators for functionals defined on a class of spectral densities. We define the concepts of H0- 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. We prove that \(\Phi (\hat \theta _T ),{\text{ where }}\hat \theta _T \) is a suitable sequence of T1/2-consistent estimators of unknown spectral density θ(λ), is H0- and IK-asymptotically efficient estimator for a nonlinear smooth functional Φ(θ).