Open Access
Translator Disclaimer
October, 1995 Adaptive Root $n$ Estimates of Integrated Squared Density Derivatives
Tiee-Jian Wu
Ann. Statist. 23(5): 1474-1495 (October, 1995). DOI: 10.1214/aos/1176324308


Based on a random sample of size $n$ from an unknown density $f$ on the real line, the nonparametric estimation of $\theta_k = \int\{f^{(k)}(x)\}^2 dx, k = 0, 1,\ldots$, is considered. These functionals are important in a number of contexts. The proposed estimates of $\theta_k$ is constructed in the frequency domain by using the sample characteristic function. It is known that the sample characteristic function at high frequency is dominated by sample variation and does not contain much information about $f$. Hence, the variation of the estimate can be reduced by modifying the sample characteristic function beyond some cutoff frequency. It is proposed to select adaptively the cutoff frequency by a generalization of the (smoothed) cross-validation. The exact convergence rate of the proposed estimate to $\theta_k$ is established. It depends solely on the smoothness of $f$. For sufficiently smooth $f$, it is shown that the proposed estimate is asymptotically normal, attains the optimal $O_p(n^{-1/2})$ rate and achieves the information bound. Finally, to improve the performance of the proposed estimate at small to moderately large $n$, two modifications are proposed. One modification is for estimating $\theta_0$; it reduces bias of the estimate. The other modification is for estimating $\theta_k, k \geq 1$; it reduces sample variation of the estimate. In simulation studies the superior performance of the proposed procedures is clearly demonstrated.


Download Citation

Tiee-Jian Wu. "Adaptive Root $n$ Estimates of Integrated Squared Density Derivatives." Ann. Statist. 23 (5) 1474 - 1495, October, 1995.


Published: October, 1995
First available in Project Euclid: 11 April 2007

zbMATH: 0843.62036
MathSciNet: MR1370292
Digital Object Identifier: 10.1214/aos/1176324308

Primary: 62G05
Secondary: 62G20

Keywords: Bandwidth selection , Characteristic function , convergence rate , Density derivative , kernel estimate , nonparametric information bound , smoothed cross-validation

Rights: Copyright © 1995 Institute of Mathematical Statistics


Vol.23 • No. 5 • October, 1995
Back to Top