PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
Tytuł artykułu

Fractional lower order covariance based-estimator for Ornstein-Uhlenbeck process with stable distribution

Identyfikatory
Warianty tytułu
PL
Estymator bazujący na ułamkowych momentach dla procesu Ornsteina-Uhlenbecka z rozkładem stabilnym
Języki publikacji
EN
Abstrakty
EN
The Ornstein-Uhlenbeck model is one of the most popular stochastic processes. It has found many interesting applications including physical phenomena. However, for many real data, the classical Ornstein-Uhlenbeck process cannot be applied. It is related to the fact that for many phenomena the vectors of observations exhibit the so-called heavy-tailed behaviour. In such cases, the modifications of the classical models need to be used. In this paper, we analyze the Ornstein-Uhlenbeck process based on stable distribution. This distribution is one of the most classical members of the heavy-tailed class of distributions. In the literature, one can find various applications of stable processes. However, the heavy-tailed property implies that the classical methods of estimation and statistical investigation cannot be applied. In this paper, we propose a new method of estimation of the stable Ornstein-Uhlenbeck process. This technique is based on the alternative measure of dependence, called fractional lower order covariance, which replaces the classical covariance for infinite-variance distribution. The proposed research is a continuation of the authors' previous studies, where the measure called covariation was proposed as the base for the estimation technique. We introduce the stable Ornstein-Uhlenbeck process and remind its main properties. In the main part, we define the new estymator of the parameters for discrete representation of the Ornstein-Uhlenbeck process. Its effectiveness is checked by Monte Carlo simulations.
PL
Proces Ornsteina-Uhlenbecka jest jednym z najbardziej popularnych procesów stochastycznych. Znalazł on wiele ciekawych praktycznych zastosowań. Należy jednak zwrócić uwagę, że klasyczny proces Ornsteina-Uhlenbecka nie może być zastosowany dla wielu danych rzeczywistych, ponieważ często pochodzą one z rozkładów ciężko- ogonwych, dla których nie istnieje drugi moment. W takich przypadkach niezbędna jest modyfikacja klasycznego modelu z wykorzystaniem rozkładu stabilnego. Z powodu zastosowania rozkładu stabilnego niezbędne jest użycie innej metody estymacji niż bazującej na autokowariancji. Zaproponowana została nowa metoda bazująca na ułamkowych momentach. Praca jest kontynuacją wcześniej otrzymanych rezultatów dla innej alternatywnej miary zależności, kowariacji. W pracy przypomniana została definicja stabilnego procesu Ornsteina-Uhlenbecka wraz z propozycją nowych estymatorów dla parametrów tego procesu.W celu sprawdzenia ich właściwości wykonane zostały symulacje Monte Carlo.
Rocznik
Strony
269--292
Opis fizyczny
Bibliogr. 47 poz., fot., tab., wykr.
Twórcy
  • Wrocław University of Science and Technology, Faculty of Pure and Applied Mathematics, Wybrzeże Wyspiańskiego 27, PL-50-370 Wrocław
  • Wrocław University of Science and Technology, Faculty of Pure and Applied Mathematics, Wybrzeże Wyspiańskiego 27, PL-50-370 Wrocław
  • Wrocław University of Science and Technology, Faculty of Pure and Applied Mathematics, Wybrzeże Wyspiańskiego 27, PL-50-370 Wrocław
  • Wrocław University of Science and Technology, Faculty of Pure and Applied Mathematics, Wybrzeże Wyspiańskiego 27, PL-50-370 Wrocław
Bibliografia
  • [1] P. Brockwell. Continuous-time ARMA processes. Handbook of statistics, 19:249-276, 2001. doi: 10.1016/S0169-7161(01)19011-5. Cited on p. 277.
  • [2] P. J. Brockwell and R. A. Davis. Time series: theory and methods. Springer Science & Business Media, 2013. Zbl 1169.62074. Cited on p. 277.
  • [3] P. J. Brockwell and R. A. Davis. Introduction to time series and forecasting. springer, 2016. Zbl 1355.62001. Cited on p. 270.
  • [4] P. J. Brockwell and T. Marquardt. Lévy-driven and fractionally integrated ARMA processes with continuous time parameter. Statistica Sinica, pages 477-494, 2005. Zbl 1070.62068. Cited on p. 270.
  • [5] K. Burnecki and A. Weron. Fractional Lévy stable motion can model subdiffusive dynamics. Physical Review E, 82 (2):021130, 2010. doi: 10.1103/PhysRevE.82.021130. Cited on p. 270.
  • [6] K. Burnecki, J. Gajda, and G. Sikora. Stability and lack of memory of the returns of the hang seng index. Physica A: Statistical Mechanics and its Applications, 390 (18-19):3136-3146, 2011. doi: 10.1016/j.physa.2011.04.025. Cited on p. 270.
  • [7] K. Burnecki, G. Sikora, A. Weron, M. M. Tamkun, and D. Krapf. Identifying diffusive motions in single-particle trajectories on the plasma membrane via fractional time-series models. Physical Review E, 99 (1):012101, 2019. doi: 10.1103/PhysRevE.99.012101. Cited on p. 270.
  • [8] R. Davis and S. Resnick. Limit theory for the sample covariance and correlation functions of moving averages. The Annals of Statistics, pages 533-558, 1986. doi: 10.1214/aos/1176349937. Zbl 0605.62092. Cited on p. 271.
  • [9] T. Frank, A. Daffertshofer, and P. Beek. Multivariate Ornstein-Uhlenbeck processes with mean-field dependent coeffcients: Application to postural sway. Physical Review E, 63 (1):011905, 2000. doi: 10.1103/PhysRevE.63.011905. Cited on p. 270.
  • [10] C. M. Gallagher. A method for fitting stable autoregressive models using autocovariation function. Statistics & Probability Letters, 53 (4):381-390, 2001. doi: 10.1016/S0167-7152(01)00041-4. Zbl 0982.62075. Cited on pp. 273, 274, and 277.
  • [11] P. Garbaczewski and R. Olkiewicz. Ornstein-Uhlenbeck-Cauchy process. Journal of Mathematical Physics, 41 (10):6843-6860, 2000. doi: 10.1063/1.1290054. Zbl 1056.82009. Cited on p. 270.
  • [12] R. Hintze, I. Pavlyukevich, et al. Small noise asymptotics and first passage times of integrated Ornstein-Uhlenbeck processes driven by α-stable Lévy processes. Bernoulli, 20 (1):265-281, 2014. doi: 10.3150/12-BEJ485. Zbl 1309.60059. Cited on p. 270.
  • [13] I. Horenko, C. Hartmann, C. Schütte, and F. Noe. Data-based parameter estimation of generalized multidimensional Langevin processes. Physical Review E, 76 (1):016706, 2007. doi: 10.1103/PhysRevE.76.016706. Cited on p. 277.
  • [14] Y. Hu and H. Long. Parameter estimation for Ornstein-Uhlenbeck processes driven by α-stable Lévy motions. Communications on Stochastic Analysis, 1 (2):1, 2007. doi: 10.31390/cosa.1.2.01. Zbl 10.1016/j.spa.2008.12.006. Cited on p. 271.
  • [15] Y. Hu and H. Long. Least squares estimator for Ornstein-Uhlenbeck processes driven by α-stable motions. Stochastic Processes and their applications, 119 (8):2465-2480, 2009. doi: 10.1016/j.spa.2008.12.006. Zbl 1171.62045. Cited on p. 271.
  • [16] A. Janicki and A.Weron. Simulation and chaotic behavior of alpha-stable stochastic processes, volume 178. CRC Press, 1993. Zbl Simulation and chaotic behavior of alpha-stable stochastic processes. Cited on p. 270.
  • [17] A. Janicki, K. Podgórski, and A. Weron. Computer simulation of α-stable Ornstein-Uhlenbeck processes. In Stochastic Processes, pages 161-170. Springer, 1993. doi: 10.1007/978-1-4615-7909-0_19. Zbl 0783.60052. Cited on p. 270.
  • [18] R. Kawai and H. Masuda. Infinite variation tempered stable Ornstein-Uhlenbeck processes with discrete observations. Communications in Statistics-Simulation and Computation, 41 (1):125-139, 2012. doi: 10.1080/03610918.2011.582561. Zbl 06073004. Cited on p. 270.
  • [19] I. A. Koutrouvelis. Regression-type estimation of the parameters of stable laws. Journal of the American Statistical Association, 75 (372):918-928, 1980. doi: 10.1080/01621459.1980.10477573. Zbl 0449.62026. Cited on p. 284.
  • [20] H. A. Kramers. Brownian motion in a field of force and the diffusion model of chemical reactions. Physica, 7 (4):284-304, 1940. doi: 10.1016/S0031-8914(40)90098-2. Zbl 0061.46405. Cited on p. 269.
  • [21] P. Kruczek, A. Wyłomańska, M. Teuerle, and J. Gajda. The modified Yule-Walker method for α-stable time series models. Physica A: Statistical Mechanics and its Applications, 469:588-603, 2017. doi: 10.1016/j.physa.2016.11.037. Zbl 1400.62185. Cited on pp. 271, 274, 277, and 284.
  • [22] T.-H. Liu and J. M. Mendel. A subspace-based direction finding algorithm using fractional lower order statistics. IEEE Transactions on Signal Processing, 49 (8):1605-1613, 2001. doi: 10.1109/78.934131. Zbl 1369.94212. Cited on p. 271.
  • [23] X. Ma and C. L. Nikias. Joint estimation of time delay and frequency delay in impulsive noise using fractional lower order statistics. IEEE Transactions on Signal Processing, 44 (11):2669-2687, 1996. doi: 10.1109/78.542175. Cited on pp. 270, 275, and 276.
  • [24] M. Maejima, K. Yamamoto, et al. Long-memory stable Ornstein-Uhlenbeck processes. Electronic Journal of Probability, 8, 2003. doi: 10.1214/EJP.v8-168. Zbl 1087.60034. Cited on p. 270.
  • [25] M. Magdziarz and K. Weron. Anomalous diffusion schemes underlying the Cole-Cole relaxation: the role of the inverse-time α-stable subordinator. Physica A: Statistical Mechanics and its Applications, 367:1-6, 2006. doi: 10.1016/j.physa.2005.12.011. Cited on p. 270.
  • [26] J. H. McCulloch. Simple consistent estimators of stable distribution parameters. Communications in Statistics-Simulation and Computation, 15 (4):1109-1136, 1986. doi: 10.1080/03610918608812563. Zbl 0612.62028. Cited on p. 284.
  • [27] D. W.-C. Miao. Analysis of the discrete Ornstein-Uhlenbeck process caused by the tick size effect. Journal of Applied Probability, 50 (4):1102-1116, 2013. doi: doi.org/10.1239/jap/138937010. Zbl Analysis of the discrete Ornstein-Uhlenbeck process caused by the tick size effect. Cited on p. 277.
  • [28] T. Mikosch, T. Gadrich, C. Kluppelberg, and R. J. Adler. Parameter estimation for ARMA models with infinite variance innovations. The Annals of Statistics, 23 (1):305-326, 1995. doi: 10.1214/aos/1176324469. Zbl 0822.62076. Cited on pp. 271 and 284.
  • [29] S. Mittnik and S. T. Rachev. Modeling asset returns with alternative stable distributions. Econometric Reviews, 12 (3):261-330, 1993. doi: 10.1080/07474939308800266. Zbl 0801.62096. Cited on p. 270.
  • [30] M. Niemann, T. Laubrich, E. Olbrich, and H. Kantz. Usage of the Mori-Zwanzig method in time series analysis. Physical Review E, 77 (1):011117, 2008. doi: 10.1103/PhysRevE.77.011117. Cited on p. 277.
  • [31] C. L. Nikias and M. Shao. Signal processing with alpha-stable distributions and applications. Wiley-Interscience, 1995. Cited on p. 270.
  • [32] J. P. Nolan. Modeling financial data with stable distributions. Handbook of Heavy Tailed Distributions in Finance, Handbooks in Finance: Book, 1:105-130, 2003. doi: 10.1016/B978-044450896-6.50005-4. Cited on p. 270.
  • [33] J. Obuchowski and A. Wyłomańska. Ornstein-Uhlenbeck process with non-Gaussian structure. Acta Physica Polonica B, 44 (5), 2013. doi: 10.5506/APhysPolB.44.1123. Zbl 1371.60140. Cited on pp. 270 and 277.
  • [34] M. Rupi, P. Tsakalides, E. Del Re, and C. L. Nikias. Constant modulus blind equalization based on fractional lower-order statistics. Signal Processing, 84 (5):881-894, 2004. doi: 10.1016/j.sigpro.2004.01.006. Zbl 1153.94330. Cited on p. 271.
  • [35] G. Samorodnitsky and M. Taqqu. Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Chapman and Hall, 1994. Zbl 0925.60027. Cited on pp. 270, 271, 272, 273, 274, 276, and 277.
  • [36] J. Ślezak and A. Weron. From physical linear systems to discrete-time series. a guide for analysis of the sampled experimental data. Physical Review E, 91 (5):053302, 2015. doi: 10.1103/PhysRevE.91.053302. Cited on p. 277.
  • [37] B. Spagnolo, S. Spezia, L. Curcio, N. Pizzolato, A. Fiasconaro, D. Valenti, P. L. Bue, E. Peri, and S. Colazza. Noise effects in two different biological systems. The European Physical Journal B, 69 (1):133-146, 2009. doi: 10.1140/epjb/e2009-00162-y. Cited on p. 270.
  • [38] G. Terdik and W. A. Woyczynski. Rosinski measures for tempered stable and related Ornstein-Uhlenbeck processes. Probability and Mathematical Statistics, 26 (2):213, 2006. Zbl 1134.60014. Cited on p. 270.
  • [39] M. A. Thornton and M. J. Chambers. The exact discretisation of CARMA models with applications in finance. Journal of Empirical Finance, 38:739-761, 2016. doi: doi.org/10.1016/j.jempfin.2016.03.006. Cited on p. 277.
  • [40] G. E. Uhlenbeck and L. S. Ornstein. On the theory of the Brownian motion. Phys. Rev., II. Ser., 36:823-841, 1930. ISSN 0031-899X. doi: 10.1103/PhysRev.36.823. Zbl 56.1277.03. Cited on p. 269.
  • [41] O. Vasicek. An equilibrium characterization of the term structure. Journal of Financial Economics, 5 (2):177-188, 1977. doi: 10.1016/0304-405X(77)90016-2. Zbl 1372.91113. Cited on p. 270.
  • [42] B. Wade Brorsen and S. R. Yang. Maximum likelihood estimates of symmetric stable distribution parameters. Communications in Statistics-Simulation and Computation, 19 (4):1459-1464, 1990. doi: 10.1080/03610919008812928. Zbl 0850.62248. Cited on p. 284.
  • [43] A. Wyłomańska. Measures of dependence for Ornstein-Uhlenbeck processes with tempered stable distribution. Acta Physica Polonica B, 42 (10), 2011. doi: 10.5506/APhysPolB.42.2049. Zbl 1371.60084. Cited on p. 270.
  • [44] A. Wyłomańska, A. Chechkin, J. Gajda, and I. M. Sokolov. Codifference as a practical tool to measure interdependence. Physica A: Statistical Mechanics and its Applications, 421:412-429, 2015. doi: 10.5506/APhysPolB.44.1123. Zbl 1395.62286. Cited on pp. 270 and 275.
  • [45] Q. Yu, G. Shen, and M. Cao. Parameter estimation for Ornstein-Uhlenbeck processes of the second kind driven by α-stable Lévy motions. Communications in Statistics-Theory and Methods, 46 (21):10864-10878, 2017. doi: 10.1080/03610926.2016.1248786. Zbl 10.31390/cosa.1.2.01. Cited on pp. 271 and 277.
  • [46] G. Żak, A. Wyłomańska, and R. Zimroz. Periodically impulsive behavior detection in noisy observation based on generalized fractional order dependency map. Applied Acoustics, 144:31-39, 2017. doi: 10.1016/j.apacoust.2017.05.003. Cited on p. 271.
  • [47] S. Zhang and X. Zhang. A least squares estimator for discretely observed Ornstein-Uhlenbeck processes driven by symmetric α-stable motions. Annals of the Institute of Statistical Mathematics, 65 (1):89-103, 2013. doi: 10.1007/s10463-012-0362-0. Zbl 06131981. Cited on p. 271.
Uwagi
Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2020).
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-2eca6273-cabe-46a8-a87b-3451cc0dae54
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.