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Automatic methods of useful signals extraction from noise background under conditions of nonparametric uncertainty. (English. Russian original) Zbl 1215.94017
Autom. Remote Control 72, No. 2, 269-282 (2011); translation from Avtom. Telemekh. 2011, No. 2, 56-70 (2011).
Summary: Three basic techniques for random signals processing are under study: the problems of filtration, interpolation, and prediction. The last advances (including those of the author) in finding smoothing parameter (bandwidth) in the problems of nonparametric kernel estimation of unknown probability densities and their derivatives made it possible to advance further in the theory of the nonparametric estimation of signals with unknown distribution. This progress gave rise to the evolution of automatic methods for signals extraction from noise under the conditions of nonparametric uncertainty. The word “automatic” is understood in the sense that the suggested methods for processing signals depend only on the observable sample. In the article, by the simple example of the additive model, the comparison is made of the nonparametric procedures for the signals processing with the known optimal processing procedures obtained at the complete statistical information about the signals and noise distributions. The results of computer modeling show that the accuracy of nonparametric signals estimates insignificantly gives up to the accuracy of optimal estimates.

94A13 Detection theory in information and communication theory
94A12 Signal theory (characterization, reconstruction, filtering, etc.)
Full Text: DOI
[1] Rosenblatt, M., Remarks on Some Nonparametric Estimates of Density Functions, Ann. Math. Statist., 1956, vol. 27, no. 3, pp. 832–837. · Zbl 0073.14602 · doi:10.1214/aoms/1177728190
[2] Parzen, E., On Estimation of a Probability Density Function and Mode, Ann. Math. Statist., 1962, vol. 33, no. 3, pp. 1065–1076. · Zbl 0116.11302 · doi:10.1214/aoms/1177704472
[3] Masry, E. and Tjostheim, D., Nonparametric Estimation and Identification of Nonlinear ARCH Time Series, Econometric Theory, 1995, vol. 11, pp. 258–289. · Zbl 1401.62171 · doi:10.1017/S0266466600009166
[4] Dobrovidov, A.V. and Koshkin, G.M., Neparametricheskoe otsenivanie signalov (Nonparametric Signal Estimation), Moscow: Nauka, 1997. · Zbl 0886.62040
[5] Dobrovidov, A.V., Nonparametric Methods of Nonlinear Filtering of Stationary Random Sequences, Autom. Remote Control, 1983, no. 6, pp. 757–768. · Zbl 0539.93076
[6] Dobrovidov, A.V., Asymptotically -Optimal Nonparametric Procedure for Nonlinear Filtering of Stationary Sequences with Unknown Statistical Characteristics, Autom. Remote Control, 1984, no. 12, pp. 1569–1576. · Zbl 0563.93064
[7] Dobrovidov, A.V., Convergence Rates of Nonparametric Filtering Estimates in Autoregression Dynamic Systems, Autom. Remote Control, 2003, no. 1, pp. 49–64. · Zbl 1117.62489
[8] Dobrovidov, A.V. and Koshkon, G.M., Nonparametric Estimation of the Logarithmic Density Derivative of Sequences with Strong Mixing, Autom. Remote Control, 2001, no. 9, pp. 1453–1476. · Zbl 1074.93035
[9] Dobrovidov, A.V., Nonparametric Interpolation of the Markov Sequence, Autom. Remote Control, 2008, no. 1, pp. 52–60. · Zbl 1248.93167
[10] Dobrovidov, A.V. and Rud’ko, I.M., Bandwidth Selection in Nonparametric Estimator of Density Derivative by Smoothed Cross-validation Method, Autom. Remote Control, 2010, no. 2, pp. 209–224. · Zbl 05790879
[11] Härdle, W., Hart, J., Marron, J.S., and Tsybakov, A.B., Bandwidth Choice for Average Derivative Estimation, J. Am. Stat. Ass., 1992, vol. 87, no. 417, pp. 218–226. · Zbl 0781.62044
[12] Hall, P., Marron, J., and Park, B., Smoothed Cross-Validation, Probab. Theory Relat. Fields, 1992, vol. 90, pp. 1–20. · Zbl 0742.62042 · doi:10.1007/BF01205233
[13] Duong, T. and Hazelton, M.L., Cross-Validation Bandwidth Matrices for Multivariate Kernel Density Estimation, Scand. J. Statist., 2005, vol. 32, pp. 485–506. · Zbl 1089.62035 · doi:10.1111/j.1467-9469.2005.00445.x
[14] Vasil’ev, V.A., Dobrovidov, A.V., and Koshkin, G.M., Neparametricheskoe otsenivanie funktsionalov ot raspredelenii statsionarnykh posledovatel’nostei (Nonparametric Estimation of Functionals of Distributions of Stationary Sequences), Moscow: Nauka, 2004. · Zbl 1068.62042
[15] Hazen, E.M., Restorations of Components of the Multidimensional Markov Process by Observations of Its Other Components, Probl. Upravlen. Teor. Inf., 1978, vol. 7, no. 4, pp. 263–275.
[16] Liptser, L.Sh. and Shiryaev, A.N., Statistika sluchainykh protsessov (Statistics of Random Processes), Moscow: Nauka, 1974.
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