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 $d{X}_t=-\alpha \frac{X_t}{T-t}dt+d{W}_t,0\le t<T$, 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 $\frac{1}{\sqrt{|\log (T-t)|}}$, 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 \cite{kp-JVA}, respectively.