  The problem of estimating the shift (or, equivalently, the center of
symmetry) of an unknown symmetric and periodic function $f$ observed in
Gaussian white noise is considered. Using the blockwise Stein method, a
penalized profile likelihood with a data-driven penalization is introduced so
that the estimator of the center of symmetry is defined as the maximizer of the
penalized profile likelihood. This estimator has the advantage of being
independent of the functional class to which the signal $f$ si assumed to
belong and, furthermore, is shown to be semiparametrically adaptive and
efficient. Moreover, the second-order term of the risk expansion of the
proposed estimator is proved to behave at least as well as the second-order
term of the risk of the best possible estimator using monotone smoothing
filter. Under mild assumptions, this estimator is shown to be second-order
minimax sharp adaptive over the whole scale of Sobolev balls with smoothness
$\beta>1$. Thus, these results extend those of \cite{DGT}, where second-order
asymptotic minimaxity is proved for an estimator depending on the functional
class containing $f$ and $\beta\geq 2$ is required.
