Maximum likelihood estimation for the non-ergodic fractional Ornstein–Uhlenbeck process

Abstract

For the non-ergodic fractional Ornstein–Uhlenbeck (fO–U) process driven by the fractional Brownian motion, we deal with the maximum likelihood estimator (MLE) of the drift parameter, assuming that the Hurst parameter \(H\) is known and is in \([1/2, 1)\). Under this setting we compute the distribution of the MLE, and explore its distributional properties by paying attention to the influence of \(H\) and the sampling span \(T\). We also derive the asymptotic distribution of the MLE as \(T\) becomes large. It is shown that, unlike the ergodic case, the asymptotic distribution depends on \(H\). We further consider the unit root testing problem in the fO–U process and compute the powers of the test based on the MLE.