Asymptotic Unbiasedness and Consistency of Cross-Correlogram Estimators of Response Functions in Linear Continuous Systems

Ірина Петрівна Блажієвська


The estimation problem of an unknown real-valued response function of a linear continuous system is considered. We suppose that a family of zero-mean stationary Gaussian processes, which are close, in some sense, to a white noise, disturbs the system. Integral-type sample input-output cross-correlograms are taken as estimators of the response function from \begin{equation}L_{2}\left(\mathbb{R}\right)\end{equation}. The corresponding cross-correlogram estimator depends on two parameters (a parameter of a scheme of series and a length of an averaging interval) and is biased. Our aim is to investigate the properties of asymptotic unbiasedness and consistency of the estimator. The main results are obtained due to additional assumptions about the uniform Lipschitz condition for the response function, and balance conditions between the correlation functions of inputs and the parameter of the scheme of series. Properties of the Fourier transform, some properties of Fejers kernels and the Young inequality for convolutions are used to prove these facts. Both asymptotic unbiasedness and consistency in mean square sense are studied in the paper.


Response function; Sample cross-correlogram; Unbiasedness; Consistency; The Young inequality for convolutions


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