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Penalized maximum likelihood and semiparametric second-order efficiency. (English) Zbl 1091.62020

Summary: We consider the problem of estimation of a shift parameter of an unknown symmetric function in Gaussian white noise. We introduce the 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 expansions 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.

MSC:

62G05 Nonparametric estimation
62G20 Asymptotic properties of nonparametric inference
62M09 Non-Markovian processes: estimation
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