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Model autoregresyjny z szumem o podwójnym rozkładzie Pareto
Języki publikacji
Abstrakty
Time series models are a popular tool commonly used to describe time-varying phenomena. One of the most popular models is the Gaussian AR. However, when the data have outlier observations with "large" values, Gaussian models are not a good choice. We therefore abandon the assumption of normality of the data distribution and propose the AR model based on the double Pareto distribution. We introduce the estimators of the model's parameters, obtained by the maximum likelihood method. For this purpose, we use the Maclaurin series expansion and the Chebyshev polynomials expansion of the likelihood function. We compare the results with the Yule-Walker estimator in the finite variance case and with the modified Yule-Walker estimator in the infinite variance case. The accuracy of the results obtained was checked by Monte Carlo simulations.
Modele szeregów czasowych to popularne narzędzie powszechnie stosowane do modelowania zjawisk zmiennych w czasie. Najpopularniejszym modelem jest gaussowski model AR, który jest stacjonarny. Jednak gdy w danych występują obserwacje odstające o „dużych“ wartościach, modele gaussowskie nie są odpowiednim narzędziem do ich modelowania. Odchodzimy zatem od założenia o normalności rozkładu danych i proponujemy model AR oparty na podwójnym rozkładzie Pareto. Przedstawiamy estymatory parametrów modelu, uzyskane metodą największej wiarygodności. W tym celu wykorzystujemy rozwinięcie funkcji warogodności w szereg zmodyfikowanym estymatorem Yule-Walkera w przypadku nieskończonej wariancji. Poprwaność otrzymanych wyników została sprawdzona za pomocą symulacji Monte Carlo.
Wydawca
Czasopismo
Rocznik
Tom
Strony
121--141
Opis fizyczny
Bibliogr. 30 poz., wykr.
Twórcy
autor
- student Wrocław University of Science and Technology: Wrocław, PL
autor
- Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, Wybrzeże Wyspiańskiego 27, Wrocław 50-370
Bibliografia
- [1] P. L. Anderson and M. M. Meerschaert. Parameter estimation for periodically stationary time series. Journal of Time Series Analysis, 26(4): 489–518, 2005. doi: 10.1111/j.1467-9892.2005.00428.x. Cited on p. 121.
- [2] B. C. Arnold. Pareto distribution. Wiley StatsRef: Statistics Reference Online, pages 1–10, 2014. Cited on p. 123.
- [3] P. J. Brockwell and R. A. Davis. Introduction to time series and forecasting, Second Edition. Springer, 2002. Zbl 1355.62001. Cited on pp. 121, 122, 123, 125, 128, and 130.
- [4] P. J. Brockwell and R. A. Davis. Time series: theory and methods. Springer Science & Business Media, 2013. Zbl 1169.62074. Cited on p. 121.
- [5] I. Fedotenkov. A bootstrap method to test for the existence of finite moments. Journal of Nonparametric Statistics, 25(2):315–322, jun 2013. doi: 10.1080/10485252.2012.752487. Cited on p. 124.
- [6] P. Franses. Periodicity and Stochastic Trends in Economic Time Series. Oxford University Press, Oxford, 1996. Zbl 0868.62088. Cited on p. 121.
- [7] J. Gao, Z.-Y. Xu, and L.-T. Zhang. Approximating long-memory DNA sequences by short-memory process. Physica A: Statistical Mechanics and its Applications, 388(17):3475 – 3485, 2009. doi:10.1016/j.physa.2005.06.099. Cited on p. 121.
- [8] P. Giri, S. Sundar, and A. Wyłomańska. Fractional lower-order covariance (floc)-based estimation for multidimensional PAR(1) model with α−stable noise. International Journal of Advances in Engineering Sciences and Applied Mathematics, 13(2):215–235, 2021. ISSN 0975-5616. doi: 10.1007/s12572-021-00301-0. Cited on p. 121.
- [9] A. Grzesiek and A. Wyłomańska. Asymptotic behavior of the cross-dependence measures for bidimensional AR(1) model with α–stable noise. Banach Center Publications, 122:133–157, 2020. doi: 10.4064/bc122-8. Cited on p. 121.
- [10] A. Grzesiek, M. Teuerle, and A. Wyłomańska. Cross-codifference for bidimensional VAR(1) time series with infinite variance. Communications in Statistics - Simulation and Computation, pages 1–26, 2019. doi: 10.1080/03610918.2019.1670840. Cited on p. 124.
- [11] A. Grzesiek, P. Giri, S. Sundar, and A. Wyłomańska. Measures of crossdependence for bidimensional periodic AR(1) model with alpha-stable distribution. Journal of Time Series Analysis, 41(6):785–807, 2020. doi: 10.1111/jtsa.12548. Cited on p. 124.
- [12] A. C. Harvey and J. H. Stock. Continuous time autoregressive models with common stochastic trends. Journal of Economic Dynamics and Control, 12(2):365–384, 1988. ISSN 0165-1889. doi: 10.1016/0165-1889(88)90046-2. Cited on p. 121.
- [13] P. Kruczek, A. Wyłomańska, M. Teuerle, and J. Gajda. The modified Yule-Walker method for alpha-stable time series models. Physica A, and 129.
- [14] P. Kruczek, W. Żuławiński, P. Pagacz, and A. Wyłomańska. Fractional lower order covariance based-estimator for Ornstein-Uhlenbeck process with stable distribution. Mathematica Applicanda, 47(2):259–292, 2019. doi: 10.14708/ma.v47i2.6506. Cited on p. 122.
- [15] F. M. Al-Athari. Parameter estimation for the double pareto distribution. Journal of Mathematics and Statistics, 7(4):289–294, oct 2011. doi:10.3844/jmssp.2011.289.294. Cited on pp. 122, 124, and 129.
- [16] K. Maraj-Zygmąt, G. Sikora, M. Pitera, and A. Wyłomańska. Goodness of-fit test for stochastic processes using even empirical moments statistic. Chaos: An Interdisciplinary Journal of Nonlinear Science, 33(1):013128, jan 2023. doi: 10.1063/5.0111505. Cited on p. 124.
- [17] J. Mason and D. C. Handscomb. Chebyshev Polynomials. Chapman and Hall/CRC, sep 2002. doi: 10.1201/9781420036114. Cited on p. 127.
- [18] C. L. Nikias and M. Shao. Signal processing with alpha-stable distributions and applications. Wiley-Interscience, 1995. URL https://dl.acm.org/doi/10.5555/210666. Cited on p. 121.
- [19] J. Nowicka-Zagrajek and A. Wyłomańska. Measures of dependence for stable AR(1) models with time-varying coefficients. Stochastic Models, 24(1):58–70, 2008. doi: 10.1080/15326340701826906. Cited on p. 121.
- [20] M. M. Rounaghi and F. Nassir Zadeh. Investigation of market efficiency and financial stability between S&P 500 and London Stock Exchange: Monthly and yearly forecasting of time series stock returns using ARMA model. Physica A: Statistical Mechanics and its Applications, 456:10 –21, 2016. doi: 10.1016/j.physa.2016.03.006. Cited on p. 121.
- [21] W. Rudin. Principles of Mathematical Analysis. International series in pure and applied mathematics. McGraw-Hill, 1976. ISBN 9780070856134. URL https://books.google.pl/books?id=kwqzPAAACAAJ. Cited on p. 126.
- [22] 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. 121 and 124.
- [23] R. Shumway and D. Stoffer. Time Series Analysis and Its Applications With R Examples, Third Edition. Springer, 2016. Zbl 1367.62004. Cited on p. 122.
- [24] L. Trapani. Testing for (in)finite moments. Journal of Econometrics, 191(1):57–68, mar 2016. doi: 10.1016/j.jeconom.2015.08.006. Cited on p. 124.
- [25] B. Troutman. Some results in periodic autoregression. Biometrika, 66: 219–228, 1979. doi: 10.2307/2335652. Cited on p. 121.
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- [27] A. Wyłomańska and J. Gajda. Stable continuous-time autoregressive process driven by stable subordinator. Physica A: Statistical Mechanics and its Applications, 444:1012–1026, feb 2016. doi:10.1016/j.physa.2015.10.081. Cited on p. 121.
- [28] A. Wyłomańska and P. Stępniak. The maximum likelihood method for student's t-distributed autoregressive model with infinite variance. Mathematica Applicanda, 48(2), mar 2021. doi: 10.14708/ma.v48i2.7059. Cited on pp. 121 and 122.
- [29] A. Wyłomańska, J. Obuchowski, R. Zimroz, and H. Hurd. Periodic autoregressive modeling of vibration time series from planetary gearbox used in bucket wheel excavator. In F. Chaari, J. Leśkow, A. Napolitano, and A. Sanchez-Ramirez, editors, Cyclostationarity: Theory and Methods, pages 171–186. Springer International Publishing, Cham, 2014. doi:10.1007/978-3-319-04187-2_12. Cited on p. 121.
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Uwagi
PL
Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023).
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-1a48e6fe-5282-4fe1-99c6-61064be2d2d6