Estimation of integral functionals of a density and its derivatives
Estimation of integral functionals of a density and its derivatives
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DOI：
10.2307/3318586
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发表时间：
1997-06-01
期刊：
影响因子：
1.5
通讯作者：
Laurent, B
中科院分区：
文献类型：
--
作者：
Laurent, B
本文考虑一类积分型密度函数φ（f，f '，...，f（（k）），.）.给出了积分phi（f，.）从f及其导数的线性和二次泛函的有效估计出发，利用phi的Taylor展开式，构造了当f足够光滑时达到n（-1/2）阶的估计.此外，我们证明了这些估计是有效的。当n（-1/2）收敛速度不能达到和k > 0时，我们也得到了最优的收敛速度.关于二次泛函的估计，更确切地说是积分平方密度导数的估计，Bickel和Ritov已经构造了有效的估计。在这里，我们提出了一种基于投影的替代构造，这种方法似乎更自然。
We consider the problem of estimating a functional of a density of the type integral phi(f, f',..., f((k)),.). The estimation of integral phi(f,.) has already been studied by the author: starting from efficient estimators of linear and quadratic functionals of f and its derivatives and using a Taylor expansion of phi, we construct estimators which achieve the n(-1/2) rate whenever fis 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, Bickel and Ritov have already constructed efficient estimators. Here we propose an alternative construction based on projections, an approach which seems more natural.