Abstract
This paper solves the problem of usefull random signal filtering in a discrete-time nonlinear observation model. As such model, we consider the multiplicative model with nonnegative signals and noises. In contrast to standard filtering problems, our statement assumes that the distribution and equation of the useful signal are unknown. To solve the nonlinear filtering problem, the idea is to employ a generalized optimal filtering equation with a feature that the optimal estimator is expressed only through characteristics of the observed process. The role of such characteristics in the equation belongs to the logarithmic derivative of the conditional multidimensional density of observations. We find the solution of the equation using nonparametric kernel estimation methods with nonsymmetrical gamma kernel functions defined on the positive semiaxis. Moreover, we establish convergence conditions for the kernel estimator of the logarithmic derivative of the multidimensional density by dependent observations, as well as derive explicit formulas for optimal bandwidths and construct a stable (regularized) filtering estimator.
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References
Lehmann, E.L., Testing Statistical Hypotheses, New York: Wiley, 1959. Translated under the title Proverka statisticheskikh gipotez, Moscow: Nauka, 1977.
Dobrovidov, A.V., Koshkin, G.M., and Vasiliev, V.A., Non-Parametric State Space Models, Heber City: Kendrick Press, 2012.
Dobrovidov, A.V., Nonparametric Methods of Nonlinear Filtering of Stationary Random Sequences, Autom. Remote Control, 1983, vol. 44, no. 6, part 2, pp. 757–758.
Pensky, M., Empirical Bayes Estimation of a Scale Parameter, Math. Methods Statist., 1996, vol. 5, pp. 316–331.
Pensky, M., A General Approach to Nonparametric Empirical Bayes Estimation, Statist., 1997, vol. 29, pp. 61–80.
Pensky, M. and Singh, R.S., Empirical Bayes Estimation of Reliability Characteristics for an Exponential Family, Can. J. Statist., 1999, vol. 27, pp. 127–136.
Markovich, L.A., The Optimal Filtering Equation and Its Connection to Kalman Filter, Tr. XII Vseross. soveshchan. po problemam upravleniya (VSPU-2014) (Proc. XII All-Russian Meeting on Control Problems (AMCP-2014)), Moscow: Inst. Probl. Upravlen., June 16–19, 2014, pp. 1108–1113, ISBN 978-5-91450-151-5.
Dobrovidov, A.V., Stable Nonparametric Signal Filtration in Nonlinear Models, Topics in Nonparametric Statistics. Proc. First Conf. of the Int. Society for Nonparametric Statistics, New York: Springer, 2014, vol. XVI, pp. 61–74.
Bosq, S., Nonparametric Statistics for Stochastic Processes. Estimation and Prediction, New York: Springer, 1996.
Ibragimov, I.A. and Linnik, Yu.V., Nezavisimye i statsionarno svyazannye velichiny (Independent and Stationary Bound Variables), Moscow: Fizmatlit, 1965.
Vasil’ev, V.A., Dobrovidov, A.V., and Koshkin, G.M., Neparametricheskoe otsenivanie funktsionalov ot raspredelenii statsionarnykh posledovatel’nostei (Nonparametric Estimation of Functionals of Stationary Sequence Distributions), Moscow: Nauka, 2004.
Parzen, E., On Estimation of a Probability Density Function and Mode, Ann. Math. Statist., 1962, vol. 33, no. 3, p. 1065.
Bouezmarni, T. and Rombouts, J., Nonparametric Density Estimation for Multivariate Bounded Data, J. Statist. Planning Inference, 2010, vol. 140, no. 1, pp. 139–152.
Chen, S.X., Probability Density Function Estimation Using Gamma Kernels, Ann. Inst. Statist. Math., 2000, vol. 52, pp. 471–480.
Markovich, L., Gamma Kernel Estimation of Multivariate Density and Its Derivative on the Nonnegative Semi-axis by Dependent Data, arXiv:1410.2507v2, 2015.
Turlach, B.A., Bandwidth Selection in Kernel Density Estimation: A Review, Louvain-la-Neuve: CORE and Inst. de Statistique, 1993.
Devroye, L. and Györfi, L., Nonparametric Density Estimation, New York: Wiley, 1985. Translated under the title Neparametricheskoe otsenivanie plotnosti, Moscow: Mir, 1988.
Tellambura, C. and Jayalath, A.D.S., Generation of Bivariate Rayleigh and Nakagami-m Fadding Envelopes, IEEE Commun. Lett., 2000, vol. 5, pp. 170–172.
Davydov, Yu.A., On the Convergence of Distributions Induced by Stationary Random Processes, Teor. Veroyatn. Primenen., 1968, vol. XIII, no. 4, pp. 730–737.
Masry, E., Probability Density Estimation from Sampled Data, IEEE Trans. Inform. Theory, 1983, vol. IT-29, no. 5, pp. 696–709.
Brown, B.M. and Chen, S.X., Beta-Bernstein Smoothing for Regression Curves with Compact Supports, Scand. J. Statist., 1999, vol. 26, pp. 47–59.
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Original Russian Text © A.V. Dobrovidov, 2016, published in Avtomatika i Telemekhanika, 2016, No. 1, pp. 72–103.
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Dobrovidov, A.V. Regularized nonparametric filtering of signal with unknown distribution in nonlinear observation model. Autom Remote Control 77, 55–80 (2016). https://doi.org/10.1134/S0005117916010045
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DOI: https://doi.org/10.1134/S0005117916010045