A Note on Parameter Estimation in Lamperti Transformed Fractional Ornstein-Uhlenbeck Processes

Abstract

The fractional Ornstein-Uhlenbeck process is characterized by two parameters, the Hurst parameter of the fractional Brownian motion and the drift parameter. The problem of estimating the two parameters in a computationally efficient way from discrete-time data is considered in the paper. The covariance kernel of the Lamperti transformed fractional Ornstein-Uhlenbeck process in the stationary case is used in the analysis. It is shown least squares based estimation of the drift parameter does not give a consistent estimate in a general case as the number of data tends to infinity and the sampling interval tends to zero. A bias compensated estimate is therefore suggested, using an innovation variance based estimate of the Hurst parameter. The Cramér-Rao lower bound for the estimation of the parameters is evaluated and the properties of the suggested estimators are illustrated numerically.