On the rate of convergence of the maximum likelihood estimator in Brownian semimartingale models

Abstract

In this paper we present a unified approach to obtaining rates of convergence for the maximum likelihood estimator (MLE) in Brownian semimartingale models of the form $$d{X}_t={\beta }_t^{n,\theta }dt+{\sigma }_t^{n}d{W}_t,t\le T_n.$$ We show that the rate of the MLE is determined by (an appropriate version of) the entropy of the parameter space with respect to the random metric h_n, defined by $$h_n^2(\theta ,\psi )={\int }_0^{T_n}{\left(\frac{{\beta }_s^{n,\theta }-{\beta }_s^{n,\psi }}{{\sigma }_s^n}\right)}^{2}ds.$$ Several known results for the rates in certain popular sub-models of the Brownian semimartingale model are shown to be special cases in our general framework.