Adaptive Root $n$ Estimates of Integrated Squared Density Derivatives
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密度导数平方积分的自适应平方根样本量估计
- 研究问题
- 如何非参数估计未知密度及其各阶导数的平方积分，并根据密度光滑程度自适应选择截断频率以控制估计误差？
- 方法与思路
- 作者利用样本特征函数在频域构造估计，以推广的平滑交叉验证选择截断频率，并针对零阶与高阶泛函分别提出有限样本修正。
- 材料陈述的贡献
- 摘要称，确定依赖密度光滑度的收敛速度；充分光滑时估计渐近正态、达到平方根样本量速度与信息界，模拟显示所提程序的优势。
- 阅读关注点
- 应核对各阶泛函所需的光滑度、截断选择程序、信息界的适用条件，以及两种修正各自减少偏差或样本变异的机制。
- 阅读边界
- 仅据版本未注明的索引摘要，不能认为所有密度均可达到平方根样本量速度；摘要未列出光滑度阈值、模拟基准或优势幅度。
原文摘要
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.