Optimal Berry-Esséen bound for Maximum likelihood estimation of the drift parameter in α-Brownian bridge
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Let T>0,α>1/2. In the present paper we consider the α-Brownian bridge defined as dX_t=-αX_t/T-tdt+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 α 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 <cit.>, respectively.
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