  Consider an estimation of the Hurst parameter $H\in(0,1)$ and the volatility
parameter $\sigma>0$ for a fractional Brownian motion with a drift term under
high-frequency observations with a finite time interval. In the present paper,
we propose a consistent estimator of the parameter $\theta=(H,\sigma)$
combining the ideas of a quasi-likelihood function based on a local Gaussian
approximation of a high-frequently observed time series and its
frequency-domain approximation. Moreover, we prove an asymptotic normality
property of the proposed estimator for all $H\in(0,1)$ when the drift process
is constant.
