Electronic Journal of Statistics

Hypothesis testing of the drift parameter sign for fractional Ornstein–Uhlenbeck process

Alexander Kukush, Yuliya Mishura, and Kostiantyn Ralchenko

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We consider the fractional Ornstein–Uhlenbeck process with an unknown drift parameter and known Hurst parameter $H$. We propose a new method to test the hypothesis of the sign of the parameter and prove the consistency of the test. Contrary to the previous works, our approach is applicable for all $H\in(0,1)$.

Article information

Electron. J. Statist., Volume 11, Number 1 (2017), 385-400.

Received: September 2016
First available in Project Euclid: 13 February 2017

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Zentralblatt MATH identifier

Primary: 60G22: Fractional processes, including fractional Brownian motion 62F03: Hypothesis testing 62F05: Asymptotic properties of tests

Fractional Brownian motion fractional Ornstein–Uhlenbeck process hypothesis testing

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Kukush, Alexander; Mishura, Yuliya; Ralchenko, Kostiantyn. Hypothesis testing of the drift parameter sign for fractional Ornstein–Uhlenbeck process. Electron. J. Statist. 11 (2017), no. 1, 385--400. doi:10.1214/17-EJS1237. https://projecteuclid.org/euclid.ejs/1486976417

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  • [1] Belfadli, R., Es-Sebaiy, K. and Ouknine, Y. (2011). Parameter estimation for fractional Ornstein–Uhlenbeck processes: non-ergodic case., Frontiers in Science and Engineering 1 1-16.
  • [2] Bertin, K., Torres, S. and Tudor, C. A. (2011). Maximum-likelihood estimators and random walks in long memory models., Statistics 45 361–374.
  • [3] Brouste, A. and Iacus, S. M. (2013). Parameter estimation for the discretely observed fractional Ornstein–Uhlenbeck process and the Yuima R package., Computational Statistics 28 1529–1547.
  • [4] Cénac, P. and Es-Sebaiy, K. (2015). Almost sure central limit theorems for random ratios and applications to LSE for fractional Ornstein-Uhlenbeck processes., Probab. Math. Statist. 35 285–300.
  • [5] Cheridito, P., Kawaguchi, H. and Maejima, M. (2003). Fractional Ornstein–Uhlenbeck processes., Electron. J. Probab. 8.
  • [6] El Machkouri, M., Es-Sebaiy, K. and Ouknine, Y. (2016). Least squares estimator for non-ergodic Ornstein–Uhlenbeck processes driven by Gaussian processes., J. Korean Stat. Soc. 45 329–341.
  • [7] El Onsy, B., Es-Sebaiy, K. and Viens, F. (2015). Parameter estimation for a partially observed Ornstein–Uhlenbeck process with long-memory noises., arXiv preprint arXiv:1501.04972.
  • [8] Es-Sebaiy, K. (2013). Berry-Esséen bounds for the least squares estimator for discretely observed fractional Ornstein–Uhlenbeck processes., Stat. Probab. Lett. 83 2372–2385.
  • [9] Es-sebaiy, K. and Ndiaye, D. (2014). On drift estimation for non-ergodic fractional Ornstein–Uhlenbeck process with discrete observations., Afr. Stat. 9 615–625.
  • [10] Es-Sebaiy, K. and Viens, F. (2016). Optimal rates for parameter estimation of stationary Gaussian processes., arXiv preprint arXiv:1603.04542.
  • [11] Hu, Y. and Nualart, D. (2010). Parameter estimation for fractional Ornstein-Uhlenbeck processes., Stat. Probab. Lett. 80 1030–1038.
  • [12] Hu, Y. and Song, J. (2013). Parameter estimation for fractional Ornstein-Uhlenbeck processes with discrete observations. In, Malliavin calculus and stochastic analysis. A Festschrift in honor of David Nualart. New York, NY: Springer 427–442.
  • [13] Kleptsyna, M. L. and Le Breton, A. (2002). Statistical analysis of the fractional Ornstein–Uhlenbeck type process., Stat. Inference Stoch. Process. 5 229–248.
  • [14] Kozachenko, Y., Melnikov, A. and Mishura, Y. (2015). On drift parameter estimation in models with fractional Brownian motion., Statistics 49 35–62.
  • [15] Kubilius, K., Mishura, Y., Ralchenko, K. and Seleznjev, O. (2015). Consistency of the drift parameter estimator for the discretized fractional Ornstein–Uhlenbeck process with Hurst index $H\in(0,\frac12)$., Electron. J. Stat. 9 1799–1825.
  • [16] Moers, M. (2012). Hypothesis testing in a fractional Ornstein-Uhlenbeck model., Int. J. Stoch. Anal. Art. ID 268568, 23.
  • [17] Samuelson, P. A. (1965). Rational theory of warrant pricing., Industrial Management Review 6 13–32.
  • [18] Tanaka, K. (2013). Distributions of the maximum likelihood and minimum contrast estimators associated with the fractional Ornstein–Uhlenbeck process., Stat. Inference Stoch. Process. 16 173–192.
  • [19] Tanaka, K. (2015). Maximum likelihood estimation for the non-ergodic fractional Ornstein–Uhlenbeck process., Stat. Inference Stoch. Process. 18 315–332.
  • [20] Tudor, C. A. and Viens, F. G. (2007). Statistical aspects of the fractional stochastic calculus., Ann. Stat. 35 1183–1212.
  • [21] Xiao, W., Zhang, W. and Xu, W. (2011). Parameter estimation for fractional Ornstein–Uhlenbeck processes at discrete observation., Appl. Math. Modelling 35 4196–4207.