In this article, we consider the drift parameter estimation problem for the nonergodic OrnsteinUhlenbeck process defined as d
X_{t}=
θX_{t}d
t + d
G_{t},
t ≥ 0 with an unknown parameter
θ > 0, where G is a Gaussian process. We assume that the process {
X_{t},
t ≥ 0} is observed at discrete time instants
t_{1}=△
_{n},…,
t_{n}=
n△
_{n}, and we construct two least squares type estimators
and
for
θ on the basis of the discrete observations {
X_{ti},
i=1,…,
n} as
n → ∞. Then, we provide sufficient conditions, based on properties of G, which ensure that
and
are strongly consistent and the sequences √
n△_{n}(

θ) and √
n△_{n}(

θ) are tight. Our approach offers an elementary proof of[11], which studied the case when
G is a fractional Brownian motion with Hurst parameter
H ∈ (1/2, 1). As such, our results extend the recent findings by[11] to the case of general Hurst parameter
H ∈ (0, 1). We also apply our approach to study subfractional OrnsteinUhlenbeck and bifractional OrnsteinUhlenbeck processes.