Distributions of the Maximum Likelihood and Minimum Contrast Estimators Associated with the Fractional Ornstein-Uhlenbeck Process

We discuss some inference problems associated with the fractional Ornstein-Uhlenbeck (fO-U) process driven by the fractional Brownian motion (fBm). In particular, we are concerned with the estimation of the drift parameter, assuming that the Hurst parameter H is known and is in [1/2, 1). Under this setting we compute the distributions of the maximum likelihood estimator (MLE) and the minimum contrast estimator (MCE) for the drift parameter, and explore their distributional properties by paying attention to the influence ofH and the sampling spanM . We shall also derive the asymptotic distributions of the two estimators as M becomes large. We further deal with the ordinary least squares estimator (OLSE) and examine the asymptotic relative efficiency. It is shown that the MCE is asymptotically efficient, while the OLSE is inefficient. We also consider the unit root testing problem in the fO-U process and compute the power of the tests based on the MLE and MCE.