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Asymptotic properties of spectrum estimate of stationary Gaussian processes. (English. Russian original) Zbl 0853.62073

J. Contemp. Math. Anal., Armen. Acad. Sci. 30, No. 1, 1-16 (1995); translation from Izv. Nats. Akad. Nauk Armen., Mat. 30, No. 1, 3-20 (1995).
Summary: Let \(X(u)\), \(u \in U\), be a zero mean real-valued stationary Gaussian process possessing a spectral density \(f(\lambda)\). We consider the general problem of nonparametric statistical estimation of spectral averages \(L(f) = \int \varphi (\lambda) f (\lambda) d \lambda\), on the basis of a sample \(\{X(u),\;0 \leq u \leq T\}\). As an estimator of \(L(f)\) we take the statistics \(L_T = \int \varphi (\lambda) I_T (\lambda) d \lambda\), where \(I_T (\lambda)\) is the periodogram of the process \(X(u)\). For spectral densities from different functional classes we prove in succession that the estimator \(L_T\) is asymptotically unbiased, mean square consistent, asymptotically normal and an asymptotically efficient estimator for \(L(f)\). Both continuous and discrete parameter cases are treated.

MSC:

62M15 Inference from stochastic processes and spectral analysis
62G20 Asymptotic properties of nonparametric inference
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