Polynomial splines and nonparametric regression
Polynomial splines and nonparametric regression
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DOI：
10.1080/10485259108832516
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发表时间：
1991
影响因子：
1.2
通讯作者：
Hung Chen
中科院分区：
文献类型：
--
作者：
Hung Chen
令 (X, Y) ∈ [0, 1]d × R 为随机向量，并令给定 X = x 的 Y 条件分布具有均值 θ(x) 并满足适当的矩条件。假设 X 的密度函数在 [0, 1]d 上远离零和无穷大。假设已知 θ(x) 是仅 x 的一般 d 维平滑函数。考虑 θ 的估计量，其形式为多项式样条，在 [0, 1]d 上的等距网格处具有简单结，其中系数是通过基于 (X, Y) 分布中大小为 n 的随机样本的最小二乘法确定的。结果表明，该估计器分别在 L 2 范数和超范数下实现了 Stone [14] 中定义的非参数回归估计的最佳收敛速度。
Let (X, Y) ∈ [0, 1]d × R be a random vector and let the conditional distribution of Y given X = x have mean θ(x) and satisfy an appropriate moment condition. It is assumed that the density function of X is bounded away from zero and infinity on [0, 1]d. Suppose that θ(x) is known to be a general d-dimensional smooth function of x only. Consider an estimator of θ having the form of a polynomial spline with simple knots at equally spaced grids over [0, 1]d where the coefficients are determined by the method of least-squares based on a random sample of size n from the distribution of (X, Y). It is shown that this estimator achieves the optimal rates of convergence for nonparametric regression estimation as defined in Stone [14] under L 2 norm and supnorm, respectively.