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.

References
- Ulf Grenander,
*A prediction problem in game theory*, Ark. Mat. **3** (1957), 371–379. MR **90486**, DOI https://doi.org/10.1007/BF02589429
- Jürgen Franke,
*Minimax-robust prediction of discrete time series*, Z. Wahrsch. Verw. Gebiete **68** (1985), no. 3, 337–364. MR **771471**, DOI https://doi.org/10.1007/BF00532645
- Jürgen Franke and H. Vincent Poor,
*Minimax-robust filtering and finite-length robust predictors*, Robust and nonlinear time series analysis (Heidelberg, 1983) Lect. Notes Stat., vol. 26, Springer, New York, 1984, pp. 87–126. MR **786305**, DOI https://doi.org/10.1007/978-1-4615-7821-5_6
- Kari Karhunen,
*Über lineare Methoden in der Wahrscheinlichkeitsrechnung*, Ann. Acad. Sci. Fennicae Ser. A. I. Math.-Phys. **1947** (1947), no. 37, 79 (German). MR **23013**
- A. N. Kolmogorov,
*Teoriya veroyatnosteĭ i matematicheskaya statistika. Izbrannye trudy*, “Nauka”, Moscow, 1986 (Russian). Edited by Yu. V. Prokhorov; With commentaries. MR **861120**
- M. Luz,
*Wiener–Kolmogorov filters for prediction of stationary processes*, Visnyk Kyiv Univ. Ser. Mat. Meh., **25** (2011), 26–29. (Ukrainian)
- M. P. Moklyachuk,
*Stochastic autoregressive sequences and minimax interpolation*, Teor. Ĭmovīr. Mat. Stat. **48** (1993), 135–146 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. **48** (1994), 95–103. MR **1445234**, DOI https://doi.org/10.1090/S0094-9000-2016-00988-7
- M. P. Moklyachuk,
*Robust procedures in time series analysis*, Theory Stoch. Process. **6(22)** (2000), no. 3–4, 127–147.
- M. P. Moklyachuk,
*Game theory and convex optimization methods in robust estimation problems*, Theory Stoch. Process. **7(23)** (2001), no. 1–2, 253–264.
- M. P. Moklyachuk,
*Robust Estimators of Functionals of Stochastic Processes*, Kyiv University Publishing House, Kyiv, 2008. (Ukrainian)
- M. S. Pinsker and A. M. Yaglom,
*On linear extrapolation of random processes with stationary $n$th increments*, Doklady Akad. Nauk SSSR (N.S.) **94** (1954), 385–388 (Russian). MR **0061308**
- M. S. Pinsker,
*The theory of curves in Hilbert space with stationary $n$th increments*, Izv. Akad. Nauk SSSR. Ser. Mat. **19** (1955), 319–344 (Russian). MR **0073957**
- B. N. Pshenichnyĭ,
*Neobkhodimye usloviya èkstremuma*, Optimizatsiya i Issledovanie Operatsiĭ. [Optimization and Operations Research], “Nauka”, Moscow, 1982 (Russian). MR **686452**
- Yu. A. Rozanov,
*Statsionarnye sluchaĭ nye protsessy*, 2nd ed., Teoriya Veroyatnosteĭ i Matematicheskaya Statistika [Probability Theory and Mathematical Statistics], vol. 42, “Nauka”, Moscow, 1990 (Russian). MR **1090826**
- Norbert Wiener,
*Extrapolation, Interpolation, and Smoothing of Stationary Time Series. With Engineering Applications*, The Technology Press of the Massachusetts Institute of Technology, Cambridge, Mass; John Wiley & Sons, Inc., New York, N. Y.; Chapman & Hall, Ltd., London, 1949. MR **0031213**
- A. M. Yaglom,
*Correlation theory of stationary and related random functions. Vol. I*, Springer Series in Statistics, Springer-Verlag, New York, 1987. Basic results. MR **893393**
- A. M. Yaglom,
*Correlation theory of stationary and related random functions. Vol. II*, Springer Series in Statistics, Springer-Verlag, New York, 1987. Supplementary notes and references. MR **915557**
- A. M. Yaglom,
*Correlation theory of processes with random stationary $n$th increments*, Mat. Sb. N.S. **37(79)** (1955), 141–196 (Russian). MR **0071672**
- A. M. Yaglom,
*Certain types of random fields in $n$-dimensional space similar to stationary stochastic processes*, Teor. Veroyatnost. i Primenen **2** (1957), 292–338 (Russian, with English summary). MR **0094844**

References
- U. Grenander,
*A prediction problem in game theory*, Ark. Mat. **3** (1957), 371–379. MR **0090486 (19:822g)**
- J. Franke,
*Minimax robust prediction of discrete time series*, Z. Wahrsch. verw. Gebiete **68** (1985), 337–364. MR **771471 (86f:62164)**
- J. Franke and H. V. Poor,
*Minimax-robust filtering and finite-length robust predictors*, Robust and Nonlinear Time Series Analysis, Lecture Notes in Statistics, Springer-Verlag, vol. 26, 1984, pp. 87–126. MR **786305 (86i:93058)**
- K. Karhunen,
*Über lineare Methoden in der Wahrscheinlichkeitsrechnung*, Ann. Acad. Sci. Fennicae, Ser. A I (1947), no. 37, 1–79. MR **0023013 (9:292i)**
- A. N. Kolmogorov,
*Probability Theory and Mathematical Statistics*, Selected works, “Nauka”, Moscow, 1986. (Russian) MR **861120 (88d:01043)**
- M. Luz,
*Wiener–Kolmogorov filters for prediction of stationary processes*, Visnyk Kyiv Univ. Ser. Mat. Meh., **25** (2011), 26–29. (Ukrainian)
- M. P. Moklyachuk,
*Stochastic autoregressive sequences and minimax interpolation*, Theory Probab. Math. Stat. **48** (1994), 95–103. MR **1445234 (98b:60081)**
- M. P. Moklyachuk,
*Robust procedures in time series analysis*, Theory Stoch. Process. **6(22)** (2000), no. 3–4, 127–147.
- M. P. Moklyachuk,
*Game theory and convex optimization methods in robust estimation problems*, Theory Stoch. Process. **7(23)** (2001), no. 1–2, 253–264.
- M. P. Moklyachuk,
*Robust Estimators of Functionals of Stochastic Processes*, Kyiv University Publishing House, Kyiv, 2008. (Ukrainian)
- M. S. Pinsker and A. M. Yaglom,
*On linear extrapolation of random processes with stationary $n$th increments*, Doklady Akad. Nauk SSSR (N. S.) **94** (1954), no. 3, 385–388. (Russian) MR **0061308 (15:806d)**
- M. S. Pinsker,
*The theory of curves in Hilbert space with stationary $n$th increments*, Izv. Akad. Nauk SSSR. Ser. Mat. **19** (1955), no. 3, 319–344. (Russian) MR **0073957 (17:514c)**
- B. N. Pshenichnyĭ,
*Necessary conditions for an extremum*, “Nauka”, Moscow, 1982. (Russian) MR **686452 (84c:49003)**
- Yu. A. Rozanov,
*Stationary Random Processes*, 2nd edition, “Nauka”, Moscow, 1990; English transl. of 1st Russian edition, Holden-Day, Inc., San Francisco, Calif.–London–Amsterdam, 1967. MR **1090826 (92d:60046)**
- 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 (11:118j)**
- A. M. Yaglom,
*Correlation Theory of Stationary and Related Random Functions. Vol. 1: Basic Results*, Springer Series in Statistics, Springer-Verlag, New York etc., 1987. MR **893393 (89a:60105)**
- A. M. Yaglom,
*Correlation Theory of Stationary and Related Random Functions. Vol. 2: Suplementary Notes and References*, Springer Series in Statistics, Springer-Verlag, New York, 1987. MR **915557 (89a:60106)**
- A. M. Yaglom,
*Correlation theory of processes with random stationary $n$th increments*, Mat. Sb. N. S. **37(79)** (1955), no. 1, 141–196. (Russian) MR **0071672 (17:167f)**
- A. M. Yaglom,
*Certain types of random fields in $n$-dimensional space similar to stationary stochastic processes*, Teor. Veroyatnost. i Primenen. **11** (1957), no. 3, 292–337. (Russian) MR **0094844 (20:1353)**

Similar Articles

Retrieve articles in *Theory of Probability and Mathematical Statistics*
with MSC (2010):
60G10,
60G25,
60G35,
62M20,
93E10,
93E11

Retrieve articles in all journals
with MSC (2010):
60G10,
60G25,
60G35,
62M20,
93E10,
93E11

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