Efficient estimation of integral functionals of a density

作者：Béatrice Laurent · 发表于：The Annals of Statistics · 年份：1996 · DOI：10.1214/aos/1032894458 · 被引用次数：123 · 研究领域：Liver Disease Diagnosis and Treatment、Statistical Methods and Inference、Bayesian Methods and Mixture Models

We consider the problem of estimating a functional of a density of the type $\int \phi (f, \cdot)$. Starting from efficient estimators of linear and quadratic functionals of f and using a Taylor expansion of $\phi$, we build estimators that achieve the $n^{-1/2}$ rate whenever f is smooth enough. Moreover, we show that these estimators are efficient. Concerning the estimation of quadratic functionals (more precisely, of integrated squared density) Bickel and Ritov have already built efficient estimators. We propose here an alternative construction based on projections, which seems more natural.