Minimax interpolation of harmonizable sequences

Authors:
M. P. Moklyachuk and V. I. Ostapenko

Translated by:
S. Kvasko

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:
3553431

Full-text PDF Free Access

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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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References
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*Complex symmetric stable variables and processes*, Contributions to Statistics, North-Holland, Amsterdam, 1983, pp. 63–79. MR **730448**
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*Prediction of stable processes: spectral and moving average representations*, Z. Wahrsch. Verw. Gebiete **66** (1984), 593–612. MR **753815**
- I. I. Dubovets’ka, O. Yu. Masyutka, and M. P. Moklyachuk,
*Interpolation of periodically correlated stochastic sequences*, Teor. Ĭmovir. Mat. Stat. **84** (2011), 43–56; English transl in. Theory Probab. Math. Stat. **84** (2012), 43–56. MR **2857415**
- I. I. Dubovets’ka and M. P. Moklyachuk,
*On minimax estimation problems for periodically correlated stochastic processes*, Contemporary Math. Statist. **2** (2014), no. 1, 1–24.
- J. Franke,
*Minimax robust prediction of discrete time series*, Z. Wahrsch. Verw. Gebiete **68** (1985), 337–364. MR **771471**
- I. I. Golichenko and M. P. Moklyachuk,
*Estimators for Functionals of Periodically Correlated Stochastic Processes*, “Interservis”, Kyiv, 2014. (Ukrainian)
- U. Grenander,
*A prediction problem in game theory*, Ark. Mat. **3** (1957), 371–379. MR **0090486**
- E. J. Hannan,
*Multiple time series*, John Wiley & Sons, Inc., New York–London–Sydney, 1970. MR **0279952**
- Y. Hosoya,
*Harmonizable stable processes*, Z. Wahrsch. Verw. Gebiete **60** (1982), 517–533. MR **665743**
- A. D. Ioffe and V. M. Tihomirov,
*Theory of extremal problems*, Studies in Mathematics and its Applications, vol. 6, North-Holland Publishing Company, Amsterdam–New York–Oxford, 1979. MR **528295**
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*Robust techniques for signal processing: A survey*, Proc. IEEE **73** (1985), 433–481.
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*Selected works by A. N. Kolmogorov. Vol. II: Probability theory and mathematical statistics* (A. N. Shiryayev, ed.), Mathematics and Its Applications. Soviet Series, vol. 26, Kluwer Academic Publishers, Dordrecht, 1992. MR **1153022**
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*Interpolation for functionals of random sequences with stationary increments constructed from observations with the noise*, Appl. Stat. Actuar. Finance Math. **2** (2012), 131–148. (Ukrainian)
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*Interpolation of functionals of stochastic sequences with stationary increments*, Teor. Ĭmovir. Mat. Stat. **87** (2012), 105–119; English transl. in Theory Probab. Math. Stat. **87** (2013), 117–133. MR **3241450**
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*Minimax interpolation problem for random processes with stationary increments*, Stat. Optim. Inf. Comput. **3** (2015), 30–41. MR **3352740**
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*Robust procedures in time series analysis*, Theory Stoch. Process. **6** (2000), no. 3–4, 127–147.
- M. P. Moklyachuk,
*Game theory and convex optimization methods in robust estimation problems*, Theory Stoch. Process. **7** (2001), no. 1–2, 253–264.
- M. P. Moklyachuk,
*Robust Estimators for Functionals of Stochastic Processes*, “Kyiv University”, Kyiv, 2008. (Ukrainian)
- M. P. Moklyachuk,
*Nonsmooth Analysis and Optimization*, “Kyiv University”, Kyiv, 2008. (Ukrainian)
- M. P. Moklyachuk and I. I. Dubovets’ka, Minimax interpolation of periodically correlated processes, Nauk. Visnyk Uzhgorod Univ. Ser. Mat. Inform.
**23** (2012), no. 2, 51–62. (Ukrainian)
- M. P. Moklyachuk and O. Yu. Masyutka,
*Interpolation of multidimensional stationary sequences*, Teor. Ĭmovir. Mat. Stat. **73** (2005), 112–119; English transl in. Theory Probab. Math. Stat. **73** (2006), 125–133.
- M. P. Moklyachuk and O. Yu. Masyutka,
*Robust estimation problems for stochastic processes*, Theory Stoch. Process. **12** (2006), no. 3–4, 88–113. MR **2316568**
- M. Moklyachuk and O. Masyutka,
*Minimax-Robust Estimation Technique for Stationary Stochastic Processes*, LAP LAMBERT Academic Publishing, 2012.
- M. Pourahmadi,
*On minimality and interpolation of harmonizable stable processes*, SIAM J. Appl. Math. **44** (1984), no. 5, 1023–1030. MR **759712**
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*Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces*, Springer, Berlin–Heidelberg, 1970. MR **0270044**
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*An analysis of the effects of spectral uncertainty on Wiener filtering*, Automatica **28** (1983), 289–293.
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*Harmonizable stable processes on groups: spectral, ergodic and interpolation properties*, Z. Wahrsch. Verw. Gebiete **68** (1985), no. 4, 473–491. MR **772194**
- N. Wiener,
*Extrapolation, Interpolation and Smoothing of Stationary Time Series. With Engineering Applications*, M. I. T. Press, Massachusetts Institute of Technology, Cambridge, Mass., 1966. MR **0031213**
- A. M. Yaglom,
*Correlation Theory of Stationary and Related Random Functions*, vol. 1: Basic Results, Springer Series in Statistics, Springer-Verlag, New York, 1987. MR **0893393**
- A. M. Yaglom,
*Correlation Theory of Stationary and Related Random Functions*, vol. 2: Supplementary Notes and References, Springer Series in Statistics, Springer-Verlag, New York, 1987. MR **915557** *.75pc

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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

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