标题： 逐点概率收敛的广义平滑样条 显示英文标题

标题： Pointwise Convergence in Probability of General Smoothing Splines

摘要： 建立样条的收敛性可以转化为一个变分问题，该问题适合使用$\Gamma$-收敛方法进行处理。 我们考虑正则化系数随着观测数量变化的情况，即$n$，如$\lambda_n=n^{-p}$所示。 利用$\Gamma$-收敛文献中的标准定理，我们证明了广义样条模型是一致的，即估计量在概率意义下的弱收敛稍弱的意义下收敛，对于$p\leq \frac{1}{2}$来说。 在没有进一步假设的情况下，我们证明了这个速率是精确的。 这与使用希尔伯特尺度的强收敛速率不同，在这种情况下通常可以选择$p>\frac{1}{2}$。 显示英文摘要

摘要： Establishing the convergence of splines can be cast as a variational problem which is amenable to a $\Gamma$-convergence approach. We consider the case in which the regularization coefficient scales with the number of observations, $n$, as $\lambda_n=n^{-p}$. Using standard theorems from the $\Gamma$-convergence literature, we prove that the general spline model is consistent in that estimators converge in a sense slightly weaker than weak convergence in probability for $p\leq \frac{1}{2}$. Without further assumptions we show this rate is sharp. This differs from rates for strong convergence using Hilbert scales where one can often choose $p>\frac{1}{2}$.