  \noindent \textbf{Abstract}: We consider the parameter estimation problem for
the Ornstein-Uhlenbeck process $X$ driven by a fractional Ornstein-Uhlenbeck
process $V$, i.e. the pair of processes defined by the non-Markovian
continuous-time long-memory dynamics $dX_{t}=-\theta X_{t}dt+dV_{t};\ t\geq 0$,
with $dV_{t}=-\rho V_{t}dt+dB_{t}^{H};\ t\geq 0$, where $\theta >0$ and $\rho
>0$ are unknown parameters, and $B^{H}$ is a fractional Brownian motion of
Hurst index $H\in (\frac{1}{2},1)$. We study the strong consistency as well as
the asymptotic normality of the joint least squares estimator
$(\hat{\theta}_{T},\widehat{\rho }% _{T}) $ of the pair $( \theta ,\rho) $,
based either on continuous or discrete observations of $\{X_{s};\ s\in \lbrack
0,T]\}$ as the horizon $T$ increases to +$\infty $. Both cases qualify formally
as partial-hbobservation questions since $V$ is unobserved. In the latter case,
several discretization options are considered. Our proofs of asymptotic
normality based on discrete data, rely on increasingly strict restrictions on
the sampling frequency as one reduces the extent of sources of observation. The
strategy for proving the asymptotic properties is to study the case of
continuous-time observations using the Malliavin calculus, and then to exploit
the fact that each discrete-data estimator can be considered as a perturbation
of the continuous one in a mathematically precise way, despite the fact that
the implementation of the discrete-time estimators is distant from the
continuous estimator. In this sense, we contend that the continuous-time
estimator cannot be implemented in practice in any na\"ive way, and serves only
as a mathematical tool in the study of the discrete-time estimators'
asymptotics.
