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Es-Sebaiy Khalifa,Jabrane Moustaaid 한국통계학회 2021 Journal of the Korean Statistical Society Vol.50 No.2
Let T > 0, > 1 2 . In the present paper we consider the -Brownian bridge defined as dX t = − Xt T−t dt + dW t, 0 ≤ t < T , where W is a standard Brownian motion. We investigate the optimal rate of convergence to normality of the maximum likelihood estimator (MLE) for the parameter based on the continuous observation {X s, 0 ≤ s ≤ t} as t ↑ T . We prove that an optimal rate of Kolmogorov distance for central limit theorem on the MLE is given by 1 √ |log(T−t)| , as t ↑ T . First we compute an upper bound and then find a lower bound with the same speed using Corollary 1 and Corollary 2 of Kim et al. (J Multivar Anal 155:284–304, 2017b) respectively.
Least squares estimator of fractional Ornstein–Uhlenbeck processes with periodic mean
Salwa Bajja,Khalifa Es-Sebaiy,Lauri Viitasaari 한국통계학회 2017 Journal of the Korean Statistical Society Vol.46 No.4
We first study the drift parameter estimation of the fractional Ornstein–Uhlenbeck process (fOU) with periodic mean for every 1/2 < H < 1. More precisely, we extend the consistency proved in Dehling et al. (2016) for 1/2 < H < 3 /4 to the strong consistency for any 1/2 < H < 1 on the one hand, and on the other, we also discuss the asymptotic normality given in Dehling et al. (2016). 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 1/2 < H < 1.
Least squares estimator for non-ergodic Ornstein–Uhlenbeck processes driven by Gaussian processes
Mohamed El Machkouri,Khalifa Es-Sebaiy,Youssef Ouknine 한국통계학회 2016 Journal of the Korean Statistical Society Vol.45 No.3
The statistical analysis for equations driven by fractional Gaussian process (fGp) is relatively recent. The development of stochastic calculus with respect to the fGp allowed to study such models. In the present paper we consider the drift parameter estimation problem for the non-ergodic Ornstein–Uhlenbeck process defined as dXt = θXt dt + dGt , t ≥ 0 with an unknown parameter θ > 0, where G is a Gaussian process. We provide sufficient conditions, based on the properties of G, ensuring the strong consistency and the asymptotic distribution of our estimatorθt of θ based on the observation {Xs, s ∈ [0, t]} as t → ∞. Our approach offers an elementary, unifying proof of Belfadli (2011), and it allows to extend the result of Belfadli (2011) to the case when G is a fractional Brownian motion with Hurst parameter H ∈ (0, 1). We also discuss the cases of subfractional Brownian motion and bifractional Brownian motion.