Interpolation of functionals of stochastic sequences with stationary increments

Authors:
M. M. Luz and M. P. Moklyachuk

Translated by:
N. Semenov

Journal:
Theor. Probability and Math. Statist. **87** (2013), 117-133

MSC (2010):
Primary 60G10, 60G25, 60G35; Secondary 62M20, 93E10, 93E11

DOI:
https://doi.org/10.1090/S0094-9000-2014-00908-4

Published electronically:
March 21, 2014

MathSciNet review:
3241450

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The problem of optimal estimation of a linear functional \[ A_N{\xi }=\sum _{k=0}^Na(k)\xi (k)\] 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 \[ m\in \mathbb Z\setminus \{0,1,\dots ,N\} \] 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_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

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