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Interpolation of functionals of stochastic sequences with stationary increments


Authors: M. M. Luz and M. P. Moklyachuk
Translated by: N. Semenov
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 87 (2012).
Journal: Theor. Probability and Math. Statist. 87 (2013), 117-133
MSC (2010): Primary 60G10, 60G25, 60G35; Secondary 62M20, 93E10, 93E11
Published electronically: March 21, 2014
MathSciNet review: 3241450
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Abstract | References | Similar Articles | Additional Information

Abstract: The problem of optimal estimation of a linear functional

$\displaystyle A_N{\xi }=\sum _{k=0}^Na(k)\xi (k)$

that depends on unknown values of a stochastic sequence $ \{\xi (m),m\in \mathbb{Z}\}$ with stationary increments of order $ n$ by observations of the sequence at points

$\displaystyle m\in \mathbb{Z}\setminus \{0,1,\dots ,N\} $

is considered. Formulas for calculating the mean square error and spectral characteristic of the optimal linear estimator of the above functional are derived in the case where the spectral density is known. In the case where the spectral density is not known, but a set of admissible spectral densities is given, the minimax-robust approach is applied to the problem of optimal estimation of a linear functional. Formulas that determine the least favorable spectral densities and the minimax spectral characteristics are proposed for a given set of admissible spectral densities.

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Additional Information

M. M. Luz
Affiliation: Faculty for Mechanics and Mathematics, Department of Probability Theory, Statistics, and Actuarial Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue, 4E, Kiev 03127, Ukraine
Email: maksim{\textunderscore}luz@ukr.net

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

DOI: https://doi.org/10.1090/S0094-9000-2014-00908-4
Keywords: Sequence with stationary increments, robust estimator, mean square error, least favorable spectral density, minimax spectral characteristic
Received by editor(s): May 7, 2012
Published electronically: March 21, 2014
Article copyright: © Copyright 2014 American Mathematical Society