linear smoothers

In nonparametric regression, a linear smoother is an estimate $\hat{m}(x)$ of $m(x)=\mathbb{E}[Y|X=x]$ such that

$$\hat{m}(x)=\sum _{i=1}^n{\ell }_i(x)Y_i=\ell (x{)}^{T}Y,$$

for some functions ${\ell }_1(x),\dots ,{\ell }_n(x)$. This is a very broad class of estimators. Indeed, partitions and trees, kernel regression, local polynomial regression, and RKHS regression are all examples of linear smoothers.

The fact that all of these methods are “linear”, in the above sense, suggests the question of whether there are “non-linear” smoothers. The answer is yes, but they are less well-studied. See wavelets.