Berry-Esseen Bounds for Approximate Maximum Likelihood Estimators in the α-Brownian Bridge

Abstract

Let T > 0, α > 1 2 . In this work we consider the problem of estimating the drift parameter of the α-Brownian bridge defined as dXt = −α Xt T−tdt + dWt, 0 ≤ t < T , where W is a standard Brownian motion. Assume that the process X is observed equidistantly in time with the step size ∆n := T n+1 , ti = i∆n, i = 0, ..., n. We will propose two approximate maximum likelihood estimators α̂n and ᾱn for the drift parameter α based on the discrete observations Xti , i = 0, ..., n. The consistency of those estimators is studied. Explicit bounds for the Kolmogorov distance in the central limit theorem for the estimators α̂n and ᾱn are obtained.