This work is intended as a contribution to a wavelet-based adaptive estimator
of the memory parameter in the classical semi-parametric framework for Gaussian
stationary processes. In particular we introduce and develop the choice of a
data-driven optimal bandwidth. Moreover, we establish a central limit theorem
for the estimator of the memory parameter with the minimax rate of convergence
(up to a logarithm factor). The quality of the estimators are attested by
simulations