Comparisons of B-spline procedures with kernel procedures in estimating regression functions and their derivatives

Xiaoling Dou, S. Shirahata
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引用次数: 1

Abstract

There are several methods to estimate regression functions and their derivatives. Among them, B-spline procedures and kernel procedures are known to be useful. However, at present, it is not determined which procedure is better than the others. In this paper, we investigate the performance of the procedures by computer simulations. Two B-spline procedures are considered. The first one is to estimate derivatives using a different roughness penalty for each degree of the derivative d. In this procedure, the smoothing parameters and the coefficients of the B-spline functions are different for each d. The second procedure is to estimate the dth derivative just by differentiating the estimated regression function d-times. In this case, the regression function and its derivatives have a common coefficient vector of B-spline functions. Two kernel procedures are also considered. The first kernel procedure used in our simulations is constructed with the Gasser-Muller estimator and a global plug-in bandwidth selector. The second one is a local polynomial fitting with a refined bandwidth selector. As a result of our simulations, we find that B-spline procedures can give better estimates than the kernel ones in estimating regression functions. For derivatives, we also find that in B-spline methods, it is necessary to choose a different smoothing parameter (or coefficient vector) for each degree of derivative; between the two kernel methods, the Gasser-Muller procedure gives better results than the local polynomial fitting in most cases. Furthermore, the first B-spline method can still work better than the Gasser-Muller procedure in the central area of the domain of the functions. But in the boundary areas, the Gasser-Muller procedure gives more stable derivative estimates than all the other methods.
B 样条程序与核程序在估计回归函数及其导数方面的比较
有几种方法可以估算回归函数及其导数。其中,已知 B-样条程序和核程序是有用的。然而,目前还没有确定哪种程序比其他程序更好。在本文中,我们通过计算机模拟来研究这些程序的性能。本文考虑了两种 B-样条程序。第一种是对导数 d 的每个度使用不同的粗糙度惩罚来估计导数。在这种程序中,B-样条函数的平滑参数和系数对每个 d 都是不同的。在这种情况下,回归函数及其导数具有共同的 B-样条函数系数向量。我们还考虑了两种核程序。我们模拟中使用的第一个核过程是用 Gasser-Muller 估计器和全局插件带宽选择器构建的。第二个内核程序是一个局部多项式拟合程序,带有一个细化带宽选择器。模拟结果表明,在估计回归函数时,B-样条程序比核程序能给出更好的估计结果。对于导数,我们还发现,在 B-样条方法中,有必要为每一级导数选择不同的平滑参数(或系数向量);在两种核方法中,Gasser-Muller 程序在大多数情况下比局部多项式拟合的结果更好。此外,在函数域的中心区域,第一种 B-样条法仍然比 Gasser-Muller 程序更有效。但在边界区域,Gasser-Muller 程序比其他所有方法都能给出更稳定的导数估计值。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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