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April 1996 On Bickel and Ritov's conjecture about adaptive estimation of the integral of the square of density derivative
Sam Efromovich, Mark Low
Ann. Statist. 24(2): 682-686 (April 1996). DOI: 10.1214/aos/1032894459

Abstract

Bickel and Ritov suggested an optimal estimator for the integral of the square of the kth 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 $k < \beta \leq 2k + 1/4$.

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Sam Efromovich. Mark Low. "On Bickel and Ritov's conjecture about adaptive estimation of the integral of the square of density derivative." Ann. Statist. 24 (2) 682 - 686, April 1996. https://doi.org/10.1214/aos/1032894459

Information

Published: April 1996
First available in Project Euclid: 24 September 2002

zbMATH: 0859.62039
MathSciNet: MR1394982
Digital Object Identifier: 10.1214/aos/1032894459

Subjects:
Primary: 62C05
Secondary: 62E20, 62G05, 62J02, 62M99

Rights: Copyright © 1996 Institute of Mathematical Statistics

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Vol.24 • No. 2 • April 1996
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