Adaptive Root $n$ Estimates of Integrated Squared Density Derivatives

Abstract

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.