  We consider the problem of estimation of a shift parameter of an unknown
symmetric function in Gaussian white noise. We introduce a notion of
semiparametric second-order efficiency and propose estimators that are
semiparametrically efficient and second-order efficient in our model. These
estimators are of a penalized maximum likelihood type with an appropriately
chosen penalty. We argue that second-order efficiency is crucial in
semiparametric problems since only the second-order terms in asymptotic
expansion for the risk account for the behavior of the ``nonparametric
component'' of a semiparametric procedure, and they are not dramatically
smaller than the first-order terms.
