Least squares estimator of fractional Ornstein-Uhlenbeck processes with periodic mean. (English) Zbl 1377.62085
Summary: We first study the drift parameter estimation of the fractional Ornstein-Uhlenbeck process (fOU) with periodic mean for every \(\frac{1}{2}<H<1\). More precisely, we extend the consistency proved in [H. Dehling et al., Stat. Inference Stoch. Process. 20, No. 1, 1–14 (2017; Zbl 1369.62214)] for \(\frac{1}{2}<H<\frac{3}{4}\) to the strong consistency for any \(\frac{1}{2}<H<1\) on the one hand, and on the other, we also discuss the asymptotic normality given in [loc. cit.]. In the second main part of the paper, we study the strong consistency and the asymptotic normality of the fOU of the second kind with periodic mean for any \(\frac{1}{2}<H<1\).
MSC:
| 62F12 | Asymptotic properties of parametric estimators |
| 60G15 | Gaussian processes |
| 60G22 | Fractional processes, including fractional Brownian motion |
Citations:
Zbl 1369.62214Software:
YUIMAReferences:
| [1] | Alos, E.; Mazet, O.; Nualart, D., Stochastic Calculus with respect to Gaussian processes, The Annals of Probability, 766-801 (2001) · Zbl 1015.60047 |
| [2] | Azmoodeh, E.; Morlanes, G. I., Drift parameter estimation for fractional Ornstein-Uhlenbeck process of the second kind, Statistics. (2013) |
| [3] | Azmoodeh, E.; Viitasaari, L., Parameter estimation based on discrete observations of fractional Ornstein-Uhlenbeck process of the second kind, Statistical Inference for Stochastic Processes, 18, 3, 205-227 (2015) · Zbl 1325.60051 |
| [4] | Brouste, A.; Iacus, S. M., Parameter estimation for the discretely observed fractional Ornstein-Uhlenbeck process and the Yuima R package, Computational Statistics, 28, 4, 1529-1547 (2012) · Zbl 1306.65034 |
| [5] | Cheridito, P.; Kawaguchi, H.; Maejima, M., Fractional Ornstein-Uhlenbeck processes, Electronic Journal of Probability, 8, 1-14 (2003) · Zbl 1065.60033 |
| [6] | Dehling, H.; Franke, B.; Woerner, J. H.C., Estimating drift parameters in a fractional Ornstein Uhlenbeck process with periodic mean, Statistical Inference for Stochastic Processes, 1-14 (2016) · Zbl 1369.62214 |
| [7] | 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) · Zbl 1342.62024 |
| [8] | El Onsy, B.; Es-Sebaiy, K.; Viens, F., Parameter Estimation for Ornstein-Uhlenbeck driven by fractional Ornstein-Uhlenbeck processes, Stochastics, 89, 2, 431-468 (2017) · Zbl 1422.62275 |
| [9] | El Onsy, B.; Es-Sebaiy, K.; Tudor, C., Statistical analysis of the non-ergodic fractional Ornstein-Uhlenbeck process of the second kind, COSA (2017), in press |
| [10] | Es-Sebaiy, K.; Ndiaye, D., On drift estimation for discretely observed non-ergodic fractional Ornstein-Uhlenbeck processes with discrete observations, Afrika Statistika, 9, 615-625 (2014) · Zbl 1329.60103 |
| [11] | Es-Sebaiy, K. & Viens, F. (2016). Optimal rates for parameter estimation of stationary Gaussian processes. Preprint: http://arxiv.org/pdf/1603.04542.pdf; Es-Sebaiy, K. & Viens, F. (2016). Optimal rates for parameter estimation of stationary Gaussian processes. Preprint: http://arxiv.org/pdf/1603.04542.pdf |
| [12] | Hamilton, J.D. (1994). Time Series Analysis. Princeton.; Hamilton, J.D. (1994). Time Series Analysis. Princeton. · Zbl 0831.62061 |
| [13] | Hu, Y.; Nualart, D., Parameter estimation for fractional Ornstein Uhlenbeck processes, Statistics & Probability Letters, 80, 1030-1038 (2010) · Zbl 1187.62137 |
| [14] | Hu, Y.; Song, J., Parameter estimation for fractional Ornstein-Uhlenbeck processes with discrete observations, (Viens, F.; etal., Malliavin Calculus and Stochastic Analysis: A Festschrift in Honor of David Nualart (2013), Springer), 427-442 · Zbl 1268.62100 |
| [15] | Kaarakka, T.; Salminen, P., On Fractional Ornstein-Uhlenbeck process, Communications on Stochastic Analysis, 5, 121-133 (2011) · Zbl 1331.60065 |
| [16] | Kleptsyna, M.; Le Breton, A., Statistical analysis of the fractional Ornstein-Uhlenbeck type process, Statistical Inference for Stochastic Processes, 5, 229-241 (2002) · Zbl 1021.62061 |
| [17] | Kloeden, P.; Neuenkirch, A., The pathwise convergence of approximation schemes for stochastic differential equations, LMS Journal of Computation and Mathematics, 10, 235-253 (2007) · Zbl 1223.60051 |
| [18] | Nourdin, I.; Peccati, G., (Normal Approximations with Malliavin Calculus: From Stein’s Method To Universality. Normal Approximations with Malliavin Calculus: From Stein’s Method To Universality, Cambridge Tracts in Mathematics (2012), Cambridge University) · Zbl 1266.60001 |
| [19] | Nualart, D., The Malliavin Calculus and Related Topics (2006), Springer-Verlag: Springer-Verlag Berlin, 2nd edition · Zbl 1099.60003 |
| [20] | Peccati, G., Gaussian Approximations of Multiple Integrals, Electronic Communications in Probability, 12, 350-364 (2007) · Zbl 1130.60029 |
| [21] | Peccati, G.; Tudor, C. A., (Gaussian Limits for Vector-Valued Multiple Stochastic Integrals. Gaussian Limits for Vector-Valued Multiple Stochastic Integrals, Séminaire de Probabilités XXXVIII (2005)), 247-262 · Zbl 1063.60027 |
| [22] | Sottinen, T.; Viitasaari, L., Parameter Estimation for the Langevin Equation with Stationary-Increment Gaussian Noise, Statistical Inference for Stochastic Processes (2017) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
