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Stochastic Recovery of Square-Integrable Functions

Authors: Bulgakov S.A., Gorshkova V.M., Khametov V.M. Published: 14.12.2020
Published in issue: #6(93)/2020  
DOI: 10.18698/1812-3368-2020-6-4-22

 
Category: Mathematics | Chapter: Computational Mathematics  
Keywords: orthogonal functions, Fourier coefficients, observation errors, projective estimator, unbiasedness, consistency

The purpose of the study was to solve the problem of stochastic recovery of square-integrable (with respect to the Lebesgue measure) functions defined on the real line from observations with additive white Gaussian noise, for the case of discrete time. The problem is a nonparametric, i.e., infinite-dimensional, estimation problem. The study substantiates the procedure of optimal recovery, in the mean-square sense, with respect to the product of the Lebesgue measure and the Gaussian measure, and describes an algorithm for recovering such square-integrable functions. Findings of research show that the constructed procedure for nonparametric recovery of a square-integrable function gives an unbiased and consistent recovery of an unknown function. This result has not been previously described. In addition, for smooth reconstructed functions, an almost optimal reconstruction procedure is introduced and substantiated, which gives an unimprovable (in order of magnitude) estimate of the dependence of the number of orthogonal functions on the number of observations. The error of the constructed almost optimal recovery procedure in relation to the optimal recovery procedure is no more than 50 %

References

[1] Vapnik V.N., ed. Algoritmy i programmy vosstanovleniya zavisimostey [Algorithms and programs for dependence recovery]. Moscow, Nauka Publ., 1984.

[2] Vapnik V.N. Vosstanovlenie zavisimostey po empiricheskim dannym [Recovery of dependences based on empirical data]. Moscow, Nauka Publ., 1979.

[3] Darkhovskii B.S. A new approach to the stochastic recovery problem. Theory Probab. Appl., 2005, vol. 49, iss. 1, pp. 51--64. DOI: https://doi.org/10.1137/S0040585X97980853

[4] Darkhovskii B.S. On a stochastic renewal problem. Theory Probab. Appl., 1999, vol. 43, iss. 2, pp. 282--288. DOI: https://doi.org/10.1137/S0040585X9797688X

[5] Darkhovsky B.S. Stochastic recovery problem. Probl. Inf. Transm., 2008, vol. 44, no. 4, pp. 303--314. DOI: https://doi.org/10.1134/S0032946008040030

[6] Darkhovskiy B. Non asymptotic minimax estimation of functions with noisy observations. Commun. Stat. Simul. Comput., 2012, vol. 41, iss. 6, pp. 787--803. DOI: https://doi.org/10.1080/03610918.2012.625326

[7] Stratonovich R.L. Efficiency of mathematical statistics methods in problems of recovery algorithm synthesis for unknown function. Izvestiya AN SSSR. Tekhnicheskaya kibernetika, 1969, no. 1, pp. 32--46 (in Russ.).

[8] Tikhonov V.I., Kul’man N.K. Nelineynaya fil’tratsiya i kvazikogerentnyy priem signalov [Nonlinear filtering and quasicoherent signal receiving]. Moscow, Sovetskoe radio Publ., 1975.

[9] Chentsov N.N. Statistical decision rules and optimal inference. AMS, 1981

[10] Ibragimov I.A., Has’minskii R.Z. Statistical estimation. Asymptotic theory. Applications of Mathematics (Applied Probability Control Economics Information and Communication Modeling and Identification Numerical Techniques Optimization), vol. 16. New York, NY, Springer, 1981. DOI: https://doi.org/10.1007/978-1-4899-0027-2

[11] Tsybakov A.B. Introduction to nonparametric estimation. Springer Series in Statistics. New York, NY, Springer, 2009. DOI: https://doi.org/10.1007/b13794

[12] Kolmogorov A.N., Fomin S.V. Elements of the theory of functions and functional analysis. Dover, 1957.

[13] Kashin B.S., Saakyan A.A. Ortogonal’nye ryady [Orthogonal series]. Moscow, Nauka Publ., 1984.

[14] Shiryaev A.N. Probability. Graduate Texts in Mathematics, vol. 95. New York, NY, Springer, 1996. DOI: https://doi.org/10.1007/978-1-4757-2539-1

[15] Borovkov A.A. Mathematical statistics. Gordon and Breach, 1998.