Robust procedures in time series analysis (Q2732670)

Traditional methods of solution of linear extrapolation, interpolation and filtering problems for stationary stochastic sequences may be employed under the condition that spectral densities of the sequences are known exactly [see, for example, \textit{A.N. Kolmogorov}, Selected works. Vol. II: Probability theory and mathematical statistics. Edited by A.N. Shiryayev. (1992; Zbl 0743.60005); and the survey by \textit{Th. Kailath}, IEEE Trans. Inf. Theory IT-20, 146-181 (1974; Zbl 0307.93040)].NEWLINENEWLINENEWLINEIn practice, however, complete information on the spectral densities is impossible in most cases. To solve this problem, the parametric or nonparametric estimates of the unknown spectral densities are found or these densities are selected by other reasonings. Then the classical estimation method is applied provided that the estimated or selected densities are the true ones. This procedure can result in a significant increasing of the value of the error as \textit{K.S. Vastola} and \textit{H.V. Poor} [Automatica 19, 289-293 (1983; Zbl 0534.93062)] have demonstrated with the help of some examples. This is a reason to search estimates which are optimal for all densities from a certain class of the admissible spectral densities. These estimates are called minimax since they minimize the maximal value of the error. During the last two decades many investigators have been interested in minimax extrapolation, and interpolation and filtering problems for stationary stochastic sequences. A survey of results in minimax (robust) methods of data processing can be found in a paper by \textit{S.A. Kassam} and \textit{H.V. Poor} [Proc. IEEE 73, No. 3, 433-481 (1985; Zbl 0569.62084)].NEWLINENEWLINENEWLINEThe paper by \textit{U. Grenander}, Ark. Mat. 6, 371-379 (1957; Zbl 0082.13302), should be mentioned as the first one where the minimax approach to the extrapolation problem for stationary processes was proposed. \textit{J. Franke} [J. Time Ser. Anal. 5, 227-244 (1984; Zbl 0576.62090); Z. Wahrscheinlichkeitstheor. Verw. Geb. 68, 337-364 (1985; Zbl 0537.60034); Note Mat. 11, 157-175 (1991; Zbl 0804.60037)], and \textit{J. Franke} and \textit{H.V. Poor} [Lect. Notes Stat. 26, 87-126 (1984; Zbl 0569.62083)] investigated the minimax extrapolation and filtering problems for stationary sequences with the help of subdifferential calculus methods. This approach makes it possible to find equations that determine the least favorable spectral densities for various classes of densities. \textit{M.P. Moklyachuk} [Ukr. Math. J. 43, No. 1, 75-81 (1991; Zbl 0727.62091); Theory Probab. Math. Stat. 42, 113-121 (1991; Zbl 0748.62051); ibid. 41, 77-84 (1990; Zbl 0721.62092); ibid. 57, 133-141 (1998; Zbl 0940.60058); ibid. 48, 95-103 (1994; Zbl 0840.60032)] applied the convex optimization technique proposed by \textit{J. Franke} [Z. Wahrscheinlichkeitstheor. Verw. Geb. 68, 337-364 (1985; Zbl 0537.60034)] to extrapolation, interpolation and filtering estimation problems for functionals which depend on the unknown values of stationary processes and sequences.NEWLINENEWLINENEWLINEIn this paper, the estimation problems are considered for functionals which depend on the unknown values of a stationary stochastic sequence from observations with noise. Formulas are obtained for calculation of the mean square error and the spectral characteristics of the optimal estimates of the functional under the condition that the spectral densities of the signal sequence and the noise sequence are known. The least favorable spectral densities and the minimax spectral characteristics of the optimal estimate of the functional are found for concrete classes of spectral densities.