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Estimation of integral functionals of a density and its derivatives. (English) Zbl 0872.62044

Summary: We consider the problem of estimating a functional of a density of the type \(\int\varphi(f,f',\dots,f^{(k)},\cdot)\). The estimation of \(\int \varphi(f,\cdot)\) has already been studied by the author [see Ann. Stat. 24, No. 2, 659-681 (1996; Zbl 0859.62038)]: starting from efficient estimators of linear and quadratic functionals of \(f\) and its derivatives and using a Taylor expansion of \(\varphi\), we construct estimators which achieve the \(n^{-1/2}\) rate whenever \(f\) is smooth enough. Moreover, we show that these estimators are efficient. We also obtain the optimal rate of convergence when the \(n^{-1/2}\) rate is not achievable and when \(k>0\). Concerning the estimation of quadratic functionals, more precisely of integrated squared density derivatives, P. J. Bickel and Y. Ritov [Sankyā, Ser. A 50, No. 3, 381-393 (1988; Zbl 0676.62037)] have already constructed efficient estimators. Here we propose an alternative construction based on projections, an approach which seems more natural.

MSC:

62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference
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