Equivalent kernels of smoothing splines in nonparametric regression for clustered/longitudinal data
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聚类与纵向数据非参数回归中平滑样条的等价核
- 研究问题
- 存在簇内相关时，平滑样条与哪些核估计方法渐近等价，工作协方差的选择又如何影响一致性、方差及局部性？
- 方法与思路
- 作者建立平滑样条与似不相关核估计的渐近等价关系，利用伪观测迭代表示推导渐近偏差和方差，并进行模拟比较。
- 材料陈述的贡献
- 摘要称，两类估计在任意工作协方差下渐近等价；样条保持一致性，采用真实协方差时方差最小，并刻画其非局部性。
- 阅读关注点
- 应核对簇数与簇大小的渐近安排、平滑参数条件，以及最小方差的比较范围，并理解工作独立性如何改变局部性。
- 阅读边界
- 仅据 OpenAlex 索引摘要，稿件版本未注明；具体正则条件和模拟结果未提供，不能将协方差稳健性解释为无条件成立。
原文摘要
For independent data, it is well known that kernel methods and spline methods are essentially asymptotically equivalent (Silverman, 1984). However, recent work of Welsh et al. (2002) shows that the same is not true for clustered/longitudinal data. Splines and conventional kernels are different in localness and ability to account for the within‐cluster correlation. We show that a smoothing spline estimator is asymptotically equivalent to a recently proposed seemingly unrelated kernel estimator of Wang (2003) for any working covariance matrix. We show that both estimators can be obtained iteratively by applying conventional kernel or spline smoothing to pseudo‐observations. This result allows us to study the asymptotic properties of the smoothing spline estimator by deriving its asymptotic bias and variance. We show that smoothing splines are consistent for an arbitrary working covariance and have the smallest variance when assuming the true covariance. We further show that both the seemingly unrelated kernel estimator and the smoothing spline estimator are nonlocal unless working independence is assumed but have asymptotically negligible bias. Their finite sample performance is compared through simulations. Our results justify the use of efficient, non‐local estimators such as smoothing splines for clustered/longitudinal data.