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Efficient Estimation of Spectral Functionals for Continuous-Time Stationary Models

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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 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 \(\Phi(\widehat{\theta}_{T})\), where \(\widehat{\theta}_{T}\) 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.

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  1. Department of Mathematics and Statistics, Boston University, 111 Cummington Street, Boston, MA, 02215, USA

    Mamikon S. Ginovyan

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  1. Mamikon S. Ginovyan
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Correspondence to Mamikon S. Ginovyan.

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Ginovyan, M.S. Efficient Estimation of Spectral Functionals for Continuous-Time Stationary Models. Acta Appl Math 115, 233–254 (2011). https://doi.org/10.1007/s10440-011-9617-7

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