Asymptotic methods in statistical theory
Asymptotic methods in statistical theory
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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 驴(I T ), where I T =I T (驴) is the periodogram of the underlying stationary Gaussian process X(t) with an unknown spectral density 驴(驴), 驴驴驴, is H- and IK-asymptotically efficient estimator for a linear functional 驴(驴), while for a nonlinear smooth functional 驴(驴) 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 驴(驴). Exact asymptotic bounds for minimax mean square risks of estimators of linear functionals are also obtained.