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.
"Penalized maximum likelihood and semiparametric second-order efficiency." Ann. Statist. 34 (1) 169 - 201, February 2006. https://doi.org/10.1214/009053605000000895