On Bickel and Ritov's conjecture about adaptive estimation of the integral of the square of density derivative
成果类型：
Article
署名作者：
Efromovich, S; Low, M
署名单位：
University of Pennsylvania
刊物名称：
ANNALS OF STATISTICS
ISSN/ISSBN：
0090-5364
发表日期：
1996
页码：
682-686
关键词：
摘要：
Bickel and Ritov suggested an optimal estimator for the integral of the square of the hth derivative of a density when the unknown density belongs to a Lipschitz class of a given order beta. In this context optimality means that the estimate is asymptotically efficient, that is, it has the best constant and rate of risk convergence, whenever beta > 2k + 1/4, and it is rate optimal otherwise. The suggested optimal estimator crucially depends on the value of beta which is obviously unknown. Bickel and Ritov conjectured that the method of cross validation leads to a corresponding adaptive estimator which has the same optimal statistical properties as the optimal estimator based on prior knowledge of beta. We show for probability densities supported over a finite interval that when beta > 2k + 1/4 adaptation is not necessary for the construction of an asymptotically efficient estimator. On the other hand, it is not possible to construct an adaptive estimator which has the same rate of convergence as the optimal nonadaptive estimator as soon as k < beta less than or equal to 2k + 1/4.