Smoothing Spline Models with Correlated Random Errors
具有相关随机误差的平滑样条模型AI 解读 ↓
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具有相关随机误差的平滑样条模型
- 研究问题
- 当非参数回归误差存在相关性时，如何同时选择平滑程度和估计相关参数，避免独立性准则导致平滑参数偏小？
- 方法与思路
- 作者扩展 GML、GCV 和 UBR 以联合估计平滑与相关参数，并通过混合效应表示联系 BLUP、REML 及软件实现。
- 材料陈述的贡献
- 摘要称建立样条与混合效应模型的联系；所述模拟中 GML 稳定且表现较好，并展示时间序列与空间数据应用。
- 阅读关注点
- 应核对相关矩阵的参数化、联合估计准则、BLUP 与 REML 对应关系，以及支持 GML 推荐结论的模拟范围。
- 阅读边界
- 材料为版本未注明的 OpenAlex 索引摘要。GML 的优势来自摘要所述模拟，不能外推到任意相关结构；具体参数条件未说明。
原文摘要
Spline-smoothing techniques are commonly used to estimate the mean function in a nonparametric regression model. Their performances depend greatly on the choice of smoothing parameters. Many methods of selecting smoothing parameters such as generalized maximum likelihood (GML), generalized cross-validation (GCV), and unbiased risk (UBR), have been developed under the assumption of independent observations. They tend to underestimate smoothing parameters when data are correlated. In this article, I assume that observations are correlated and that the correlation matrix depends on a parsimonious set of parameters. I extend the GML, GCV, and UBR methods to estimate the smoothing parameters and the correlation parameters simultaneously. I also relate a smoothing spline model to three mixed-effects models. These relationships show that the smoothing spline estimates evaluated at design points are best linear unbiased prediction (BLUP) estimates and that the GML estimates of the smoothing parameters and the correlation parameters are restricted maximum likelihood (REML) estimates. They also provide a way to fit a spline model with correlated errors using the SAS procedure proc mixed. Simulations are conducted to evaluate and compare the performance of the GML, GCV, UBR methods and the method proposed by Diggle and Hutchinson. The GML method is recommended, because it is stable and works well in all simulations. It performs better than other methods, especially when the sample size is not large. I illustrate my methods with applications to time series data and to spatial data.