Stein Shrinkage with Penalization and Second Order Efficiency in Semiparametrics

Abstract: The problem of estimating the centre of symmetry of an unknown periodic function observed in Gaussian white noise is considered. Using the penalized blockwise Stein method, a smoothing filter allowing to define the penalized profile likelihood is proposed. The estimator of the centre of symmetry is then the maximizer of this penalized profile likelihood. This estimator is shown to be semiparametrically adaptive and efficient. Moreover, the second order term of its risk expansion is proved to behave at least as well as the second order term for 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 β > 1. Thus, these results improve on Dalalyan, Golubev and Tsybakov (2006), where β ≥ 2 is required.