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An exponential nonuniform Berry–Esseen bound of the maximum likelihood estimator in a Jacobi process

Part of: Stochastic analysis Limit theorems

Published online by Cambridge University Press:  14 February 2024

Hui Jiang ,
Qihao Lin  and
Shaochen Wang Open the ORCID record for Shaochen Wang [Opens in a new window]
Hui Jiang*
Affiliation:
Nanjing University of Aeronautics and Astronautics
Qihao Lin*
Affiliation:
Nanjing University of Aeronautics and Astronautics
Shaochen Wang*
Affiliation:
South China University of Technology
*
*Postal address: School of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China.
*Postal address: School of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China.
**Postal address: School of Mathematics, South China University of Technology, Guangzhou 510640, China. Email: mascwang@scut.edu.cn
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Abstract

We establish the exponential nonuniform Berry–Esseen bound for the maximum likelihood estimator of unknown drift parameter in an ultraspherical Jacobi process using the change of measure method and precise asymptotic analysis techniques. As applications, the optimal uniform Berry–Esseen bound and optimal Cramér-type moderate deviation for the corresponding maximum likelihood estimator are obtained.

Keywords

Berry–Esseen boundmaximum likelihood estimatorJacobi processCramér-type moderate deviation

MSC classification

Primary: 60H10: Stochastic ordinary differential equations
Secondary: 60F10: Large deviations
Type
Original Article
Information
Journal of Applied Probability , First View , pp. 1 - 18
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Ackerer, D., Filipović, D. and Pulido, S. (2018). The Jacobi stochastic volatility model. Finance Stoch. 22, 667–700.CrossRefGoogle Scholar
Ahdida, A. and Alfonsi, A. (2013). A mean-reverting SDE on correlation matrices. Stoch. Process. Appl. 129, 1472–1520.CrossRefGoogle Scholar
Ahlfors, L. V. (1979). Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable, 3rd edn. McGraw-Hill, New York.Google Scholar
Bercu, B. and Richou, A. (2015). Large deviations for the Ornstein–Uhlenbeck process with shift. Adv. Appl. Prob. 47, 880–901.CrossRefGoogle Scholar
Bernis, G. and Scotti, S. (2017). Alternative to beta coefficients in the context of diffusions. Quant. Finance 17, 275–288.CrossRefGoogle Scholar
Biane, P., Pitman, J. and Yor, M. (2001). Probability laws related to the Jacobi theta and Riemann zeta functions, and Brownian excursions. Bull. Amer. Math. Soc. 38, 435–465.CrossRefGoogle Scholar
Delbane, F. and Shirakawa, H. (2002). An interest rate model with upper and lower bound. Asia-Pacific Financial Markets 9, 191–209.CrossRefGoogle Scholar
Demni, N. and Zani, M. (2009). Large deviations for statistics of Jacobi process. Stoch. Process. Appl. 119, 518–533.CrossRefGoogle Scholar
Ethier, S. N. and Kurtz, T. G. (2005) Markov Processes: Characterizations and Convergence, 2nd edn. John Wiley, New York.Google Scholar
Fan, X. Q., Ion, G., Liu, Q. S. and Shao, Q. M. (2019). Self-normalized Cramér type moderate deviations for martingales. Bernoulli 25, 2793–2823.CrossRefGoogle Scholar
Fan, X. Q. and Shao, Q. M. (2023) Cramér’s moderate deviations for martingales with applications. Submitted.Google Scholar
Florens-Landais, D. and Pham, H. (1999). Large deviations in estimate of an Ornstein–Uhlenbeck model. J. Appl. Prob. 36, 60–77.CrossRefGoogle Scholar
Gouriéroux, C. and Jasiak, J. (2006). Multivariate Jacobi process with application to smooth transitions. J. Econometrics 131, 475–505.CrossRefGoogle Scholar
Jiang, H., Gao, F. Q. and Zhao, S. J. (2009). Moderate deviations for statistics of Jacobi process. Chinese Ann. Math. Ser. A 30, 479–490.Google Scholar
Jiang, H. and Zhou, J. Y. (2023). An exponential nonuniform Berry–Esseen bound of the fractional Ornstein–Uhlenbeck process. J. Theoret. Prob. 36, 1037–1058.CrossRefGoogle Scholar
Karlin, S. and Taylor, H. M. (1981). A Second Course in Stochastic Processes, 1st edn. Academic Press, New York.Google Scholar
Kutoyants, Y. A. (2004). Statistical Inference for Ergodic Diffusion Process, 1st edn. Springer, London.CrossRefGoogle Scholar
Petrov, V. (1975). Sums of Independent Random Variables, 1st edn. Springer, New York.Google Scholar
Wang, Q. Y. and Jing, B. Y. (1999). An exponential nonuniform Berry–Esseen bound for self-normalized sums. Ann. Prob. 27, 2068–2088.Google Scholar
Zang, Q. P. and Zhang, L. X. (2016). A general lower bound of parameter estimation for reflected Ornstein–Uhlenbeck processes. J. Appl. Prob., 53, 22–32.CrossRefGoogle Scholar