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

• Published in: Journal of the Korean Statistical Society Vol. 50; no. 2; pp. 403 - 418
• Main Authors: Es-Sebaiy, Khalifa, Moustaaid, Jabrane
• Format: Journal Article
• Language: English
• Published: Singapore Springer Singapore 01.06.2021
• Subjects: Applied Statistics; Bayesian Inference; Mathematics and Statistics; Research Article; Statistical Theory and Methods; Statistics; Statistics and Computing/Statistics Programs; Kolmogorov distance; MLE; 60H07; 62F12; Rate of convergence; Malliavin calculus; 60F05; Brownian bridge
• ISSN: 1226-3192; 2005-2863
• DOI: 10.1007/s42952-020-00084-3

Let T > 0 , α > 1 2 . In the present paper we consider the α -Brownian bridge defined as d X t = - α X t T - t d t + d W 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 Kim et al. (J Multivar Anal 155:284–304, 2017b) respectively.