Root n Bandwidth Selectors for Kernel Estimation of Density Derivatives
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密度导数核估计的根号 n 带宽选择方法
- 研究问题
- 如何为未知密度各阶导数的核估计自适应选择带宽，并在充分光滑条件下获得较快的相对收敛速度与效率？
- 方法与思路
- 作者分析积分平方偏差的核估计，分别提出基于广义平滑交叉验证与偏差展开插件法的两类自适应带宽选择器。
- 材料陈述的贡献
- 摘要称两类选择器在充分光滑条件下渐近正态，达到根号 n 相对收敛速度及猜想的信息界，并报告一、二阶导数模拟优势。
- 阅读关注点
- 应核对信息界为何仍标为猜想、不同导数阶数所需光滑性，以及偏差估计、先导带宽和交叉验证比较的实际实现。
- 阅读边界
- 仅据 OpenAlex 索引摘要，稿件版本未注明。较高阶导数需要更强光滑性，信息界被标为猜想，有限样本优势仅来自所述模拟。
原文摘要
Based on a random sample of size n from an unknown density f on the real line, the problem of adaptively selecting the bandwidth in kernel estimation of f (k) is investigated, where f (k) denotes the kth derivative of f. The estimates of density derivatives can be used to evaluate modes and inflection points of f and can be applied to the estimation of scores in certain additive models and to the empirical verification of the law of demand in econometrics. For all k, the information bounds for bandwidth selectors are given, and two types of adaptive bandwidth selectors are proposed. A careful analysis of the kernel estimate of the integrated squared bias (ISB) term, contained in a generalization of smoothed cross-validation, indicates that the “sync kernel” is the proper choice for estimating ISB and, consequently, leads to the first type of selector. The second type of selector is based on the plug-in method, which involves expanding ISB up to suitable order and estimating integrated squared density derivatives. It is shown that for all k and sufficiently smooth f and kernel, both of the proposed selectors are asymptotically normal with the optimal Op (n -1/2) relative convergence rate and achieve the (conjectured) information bound. This asymptotically optimal performance requires stronger smoothness assumptions on f and kernel for larger k. For k = 1 and 2, extensive simulation studies have been done, and the excellent performances of the proposed selectors at practical sample sizes are clearly demonstrated. In particular, the proposed selectors perform conclusively better than the one selected by cross-validation.