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Minimax interpolation of harmonizable sequences


Authors: M. P. Moklyachuk and V. I. Ostapenko
Translated by: S. Kvasko
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 92 (2015).
Journal: Theor. Probability and Math. Statist. 92 (2016), 135-146
MSC (2010): Primary 60G10, 60G25, 60G35; Secondary 62M20, 93E10, 93E11
DOI: https://doi.org/10.1090/tpms/988
Published electronically: August 10, 2016
MathSciNet review: 1445234
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Abstract | References | Similar Articles | Additional Information

Abstract: The problem of estimation of the functional $ A_N \xi =\sum _{j = 0}^{N} a_j \xi _j$ that depends on unknown values $ \xi _j$, $ j=0,1,\dots ,N$, of a harmonizable symmetric $ \alpha $-stable random sequence $ \xi _n$, $ n\in \mathbb{Z}$, by using observations of the sequence at the points $ n\in \mathbb{Z}\setminus \{0,1,\dots ,N\}$ is studied under one of the conditions, either a condition of spectral certainty or a condition of spectral uncertainty. Expressions for calculating the value of the error and spectral characteristic of the optimal linear estimator of the functional are obtained under the condition of spectral certainty in the case where the spectral density of a sequence is known. In the case of spectral uncertainty where the spectral density of a sequence is not known but a class of admissible spectral densities is given, we propose relations to determine the least favorable spectral density and the minimax spectral characteristic.


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Additional Information

M. P. Moklyachuk
Affiliation: Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, Kyiv National Taras Shevchenko University, Volodymyrs’ka Street, 64/13, 01601, Kyiv, Ukraine
Email: mmp@univ.kiev.ua

V. I. Ostapenko
Affiliation: Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, Kyiv National Taras Shevchenko University, Volodymyrs’ka Street, 64/13, 01601, Kyiv, Ukraine
Email: vt.ostapenko@gmail.com

DOI: https://doi.org/10.1090/tpms/988
Keywords: Harmonizable sequence, robust estimator, least favorable spectral density, minimax spectral characteristic
Received by editor(s): May 15, 2015
Published electronically: August 10, 2016
Article copyright: © Copyright 2016 American Mathematical Society