Optimal Berry–Esseen bound for an estimator of parameter in the Ornstein–Uhlenbeck process

• Published in: Journal of the Korean Statistical Society Vol. 46; no. 3; pp. 413 - 425
• Main Authors: Kim, Yoon Tae, Park, Hyun Suk
• Format: Journal Article
• Language: English
• Published: Singapore Elsevier B.V 01.09.2017; Springer Singapore; 한국통계학회
• Subjects: Applied Statistics; Bayesian Inference; Berry–Esseen bound; Edgeworth expansion; Malliavin calculus; Maximum likelihood estimator; Multiple stochastic integral; Ornstein–Uhlenbeck process; Statistical Theory and Methods; Statistics; Statistics and Computing/Statistics Programs; 통계학; secondary; Maximum likelihood estimator; Edgeworth expansion; Berry–Esseen bound; Multiple stochastic integral; Ornstein–Uhlenbeck process; Malliavin calculus; primary; Ornstein-Uhlenbeck process; Berry-Esseen bound; primary 60H07; secondary 60F25
• ISSN: 1226-3192; 2005-2863
• DOI: 10.1016/j.jkss.2017.01.002

This paper is concerned with the study of the rate of central limit theorem for the maximum likelihood estimator θˆT of the unknown parameter θ>0, based on the observation X={Xt,0≤t≤T}, occurring in the drift coefficient of an Ornstein–Uhlenbeck process dXt=−θXtdt+dWt,X0=0 for 0≤t≤T, where {Wt,t≥0} is a standard Brownian motion. The tool we use is an Edgeworth expansion with an explicitly expressed remainder. We prove that upper and lower bounds, obtained by controlling the remainder term, give an optimal rate 1T in Kolmogorov distance for normal approximation of θˆT.