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Moderate Deviations for Parameter Estimation in the Fractional Ornstein–Uhlenbeck Processes with Periodic Mean

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Abstract

In this paper, we study the asymptotic properties for the drift parameter estimators in the fractional Ornstein–Uhlenbeck process with periodic mean function and long range dependence. The Cremér-type moderate deviations, as well as the moderation deviation principle with explicit rate function can be obtained.

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References

  1. Bajja, S., Es-Sebaiy, K., Viitasaari, L.: Least square estimator of fractional Ornstein Uhlenbeck processes with periodic mean. Journal of the Korean Statistical Society, 46, 608–622 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bercu, B., Rouault, A.: Sharp large deviations for the Ornstein–Uhlenbeck process. Theory of Probability and Its Applications, 46, 1–19 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bercu, B., Coutin, L., Savy, N.: Sharp large deviations for the fractional Ornstein–Uhlenbeck process. Theory of Probability and Its Applications, 55, 575–610 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bercu, B., Richou, A.: Large deviations for the Ornstein–Uhlenbeck process with shift. Advances in Applied Probability, 47, 880–901 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bercu, B., Richou, A.: Large deviations for the Ornstein–Uhlenbeck process without tears. Statistics and Probability Letters, 123, 45–55 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chang, J. Y., Shao, Q. M., Zhou, W. X.: Cramér-type moderate deviations for Studentized two-sample U-statistics with applications. The Annals of Statistics, 44, 1931–1956 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen, Y., Kuang, N. H., Li, Y.: Berry–Esséen bound for the parameter estimation of fractional Ornstein–Uhlenbeck processes. Stochastics and Dynamics, 20, 2050023 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen, Y., Zhou, H. J.: Parameter estimation for an Ornstein–Uhlenbeck process driven by a general Gaussian noise. Acta Mathematica Scientia, 41, 573–595 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cheridito, P., Kawaguchi, H., Maejima, M.: Fractional Ornstein–Uhlenbeck processes. Electronic Journal of Probability, 8, Paper No. 3, 14 pp. (2003)

  10. Dehling, H., Franke, B., Woerner, J. H. C.: Estimating drift parameters in a fractional Ornstein–Ulenbeck process with periodic mean. Statistical Inference for Stochastic Processes, 20, 1–14 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  11. El Machkouri, M., Es-Sebaiy, K., Ouknine, Y.: Least squares estimator for non-ergodic Ornstein–Uhlenbeck processes driven by Gaussian processes. Journal of the Korean Statistical Society, 45, 329–341 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gao, F. Q., Jiang, H.: Deviation inequalities and moderate deviations for estimators of parameters in an Ornstein–Uhlenbeck process with linear drift. Electronic Communications in Probability, 14, 210–223 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gao, F. Q., Jiang, H.: Deviation inequalities for quadratic Wiener functionals and moderate deviations for parameter estimators. Science China Mathematics, 60, 1181–1196 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gao, F. Q., Zhao, X. Q.: Delta method in large deviations and moderate deviations for estimators. The Annals of Statistics, 39, 1211–1240 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gao, F. Q., Xiong, J., Zhao, X. Q.: Moderate deviations and nonparametric inference for monotone functions. The Annals of Statistics, 46, 1225–1254 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  16. Guillin, A., Liptser, R.: Examples of moderate deviation principles for diffusion processes. Discrete and Continuous Dynamical Systems, Series B, 6, 803–828 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hu, Y., Nualart, D.: Parameter estimation for fractional Ornstein–Uhlenbeck processes. Statistics and Probability Letters, 80, 1030–1038 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hu, Y., Nualart, D., Zhou, H.: Parameter estimation for fractional Ornstein–Uhlenbeck processes of general Hurst parameter. Statistical Inference for Stochastic Processes, 22, 111–142 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  19. Jiang, H., Liu, J. F., Wang, S. C.: Self-normalized asymptotic properties for the parameter estimation in fractional Ornstein–Uhlenbeck process. Stochatics and Dynamics, 19, 1950018 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kaarakka, T., Salminen, P.: On fractional Ornstein–Uhlenbeck process. Communications on Stochastic Analysis, 5, 121–133 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kleptsyna, M., Le Breton, A.: Statistical analysis of the fractional Ornstein–Uhlenbeck type process. Statistical Inference for Stochastic Processes, 5, 229–248 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kubilius, K., Mishura, Y., Ralchenko, K.: Consistency of the drift parameter estimator for the discretized fractional Ornstein–Uhlenbeck process with Hurst index \(H\in(0,{1\over{2}})\). Electronic Journal of Statistics, 9, 1799–1825 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  23. Lee, C., Song, J.: On drift parameter estimation for reflected fractional Ornstein–Uhlenbeck processes. Stochastics, 88, 751–778 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  24. Major, P.: On a multivariate version of Bernsteins inequality. Electronic Journal of Probability, 12, 966–988 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  25. Major, P.: Tail behavior of multiple integrals and U-statistics. Probability Surveys, 2, 448–505 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  26. Nualart, D., Peccati, G.: Central limit theorems for sequences of multiple stochastic integrals. The Annals of Probability, 33, 177–193 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  27. Nualart, D.: The Malliavin Calculus and Related Topics, Springer-Verlag, 2006

  28. Nualart, D., Ortiz-Latorre, S.: Central limit theorems for multiple stochastic integrals and Malliavin calculus. Stochastic Processes and Their Applications, 118, 614–628 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  29. Schulte, M., Thäle, C.: Cumulants on Wiener chaos: Moderate deviations and the fourth moment theorem. Journal of Functional Analysis, 270, 2223–2248 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  30. Shen, G. J., Yu, Q., Li, Y. M.: Least squares estimator of Ornstein–Uhlenbeck processes driven by fractional Lévy processes with periodic mean. Frontiers of Mathmatics in China, 14, 1281–1302 (2019)

    Article  MATH  Google Scholar 

  31. Shen, G. J., Yu, Q.: Least squares estimator for Ornstein–Uhlenbeck processes driven by fractional Lévy processes from discrete observations. Statistical Papers, 60, 2253–2271 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  32. Shen, G. J., Wang, Q. B., Yin, X. W.: Parameter estimation for the discretely observed Vasicek model with small fractional Lévy noise. Acta Mathematica Sinica, English Series, 36, 443–461 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  33. Shen, G. J., Yu, Q., Tang, Z.: The least squares estimator for an Ornstein–Uhlenbeck processes driven by a Hermite process with a periodic mean. Acta Mathematica Scientia, 41, 517–534 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  34. Shevchenko, R., Tudor, C. A.: Parameter estimation for the Rosenblatt Ornstein–Uhlenbeck process with periodic mean. Statistical Inference for Stochastic Processes, 23, 227–247 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  35. Shevchenko, R., Woerner, J.: Inference for fractional Ornstein–Uhlenbeck type processes with periodic mean in the non-ergodic case. Stochastic Analysis and Applications, 40, 589–609 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  36. Tudor, C. A., Viens, F. G.: Statistical aspects of the fractional stochastic calculus. The Annals of Statistics, 35, 1183–1212 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  37. Yu, Q.: Least squares estimator of fractional Ornstein–Uhlenbeck processes with periodic mean for general Hurst parameter. Statistical Papers, 62, 795–815 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  38. Zhao, S. J., Gao, F. Q.: Large deviations for parameter estimators of some time inhomogeneous diffusion process. Acta Mathematica Sinica, English Series, 27, 2245–2258 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

We would like to express our great gratitude to the anonymous reviewer for the careful reading and insightful comments, which surely lead to an improved presentation of this paper.

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Authors and Affiliations

  1. School of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, P. R. China

    Hui Jiang

  2. The People’s Bank of China Operations Office of Nanjing Branch, Nanjing, 210016, P. R. China

    Shi Min Li

  3. School of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou, 310018, P. R. China

    Wei Gang Wang

Authors
  1. Hui Jiang
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  2. Shi Min Li
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  3. Wei Gang Wang
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Correspondence to Wei Gang Wang.

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Conflict of Interest The authors declare no conflict of interest.

Additional information

Hui Jiang is supported by the Natural Science Foundation of Jiangsu Province of China (Grant No. BK20231435) and Fundamental Research Funds for the Central Universities (Grant No. NS2022069); Weigang WANG is supported by Natural Science Foundation of Zhejiang Province (Grant No. LY19A010004)

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Jiang, H., Li, S.M. & Wang, W.G. Moderate Deviations for Parameter Estimation in the Fractional Ornstein–Uhlenbeck Processes with Periodic Mean. Acta. Math. Sin.-English Ser. (2023). https://doi.org/10.1007/s10114-023-2157-z

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  • DOI: https://doi.org/10.1007/s10114-023-2157-z

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