  The statistical analysis for equations driven by fractional Gaussian process
(fGp) is relatively recent. The development of stochastic calculus with respect
to the fGp allowed to study such models. In the present paper we consider the
drift parameter estimation problem for the non-ergodic Ornstein-Uhlenbeck
process defined as $dX_t=\theta X_tdt+dG_t,\ t\geq0$ with an unknown parameter
$\theta>0$, where $G$ is a Gaussian process. We provide sufficient conditions,
based on the properties of $G$, ensuring the strong consistency and the
asymptotic distribution of our estimator $\widetilde{\theta}_t$ of $\theta$
based on the observation $\{X_s,\ s\in[0,t]\}$ as $t\rightarrow\infty$. Our
approach offers an elementary, unifying proof of \cite{BEO}, and it allows to
extend the result of \cite{BEO} to the case when $G$ is a fractional Brownian
motion with Hurst parameter $H\in(0,1)$. We also discuss the cases of
subfractional Brownian motion and bifractional Brownian motion.
