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Minimax interpolation of stochastic processes with stationary increments from observations with noise


Authors: M. M. Luz and M. P. Moklyachuk
Translated by: S. Kvasko
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 94 (2016).
Journal: Theor. Probability and Math. Statist. 94 (2017), 121-135
MSC (2010): Primary 60G10, 60G25, 60G35; Secondary 62M20, 93E10, 93E11
DOI: https://doi.org/10.1090/tpms/1013
Published electronically: August 25, 2017
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Abstract | References | Similar Articles | Additional Information

Abstract: The problem of optimal estimation of the linear functional

$\displaystyle A_T{\xi }=\int _0^Ta(t)\xi (t)\,dt $

depending on unknown values of the stochastic process $ \xi (t)$ with stationary increments from observations of the process $ \xi (t)+\eta (t)$ at points $ t\in \mathbb{R}\setminus [0;T]$ is considered, where $ \eta (t)$ is a stationary stochastic process uncorrelated with $ \xi (t)$. Formulas for calculating the mean square error and spectral characteristic of the optimal linear estimate of the functional are proposed in the case where spectral densities are known. Otherwise relations that determine the least favorable spectral densities and minimax spectral characteristics are proposed for given sets of admissible spectral densities.

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Additional Information

M. M. Luz
Affiliation: Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty of Mechanics and Mathematics, Taras Shevchenko National University of Kyiv, Academician Glushkov Avenue, 4E, Kyiv 03127, Ukraine
Email: maksim_luz@ukr.net

M. P. Moklyachuk
Affiliation: Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty of Mechanics and Mathematics, Taras Shevchenko National University of Kyiv, Academician Glushkov Avenue, 4E, Kyiv 03127, Ukraine
Email: mmp@univ.kiev.ua

DOI: https://doi.org/10.1090/tpms/1013
Keywords: Stochastic process with stationary increments, robust estimate, mean square error, least favorable spectral density, minimax characteristic
Received by editor(s): April 12, 2016
Published electronically: August 25, 2017
Article copyright: © Copyright 2017 American Mathematical Society