Optimal Berry-Esséen bound for maximum likelihood estimation of the drift parameter in $$\alpha $$-Brownian bridge

Abstract

Let $$T>0,\alpha >\frac{1}{2}$$ . In the present paper we consider the $$\alpha $$ -Brownian bridge defined as $$dX_t=-\alpha \frac{X_t}{T-t}dt+dW_t,\, 0\le t< T$$ , 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 $$ \alpha $$ based on the continuous observation $$\{X_s,0\le s\le t\}$$ as $$t\uparrow T$$ . We prove that an optimal rate of Kolmogorov distance for central limit theorem on the MLE is given by $$\frac{1}{\sqrt{\vert \log (T-t)\vert }}$$ , as $$t\uparrow T$$ . 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.