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I'm learning different smoothing methods and the term "effective kernel" came up and I don't really understand it.

By definition, for a smoothing method, the vector of estimates $$\hat{f}=(\hat{f_n}(x_1),\cdots,\hat{f_n}(x_n))$$ can be written as: $$\hat{f}=Sy$$ where $S$ is the smoothing matrix (or hat matrix), and the i-th row of $S$ is called the effective kernel.

What does this row mean? How does this row (or the effective kernel) change if we have different smoothing parameters (i.e. different bandwidth in kernel smoothing). Does it mean: for a range of bandwidth, the expected possible $y$ values we get?

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Note that $$\hat f_n(x) = \frac{1}{n} \sum_{i=1}^n k(x, x_i) \, y_i = \begin{bmatrix} \frac1n k(x, x_1) & \dots & \frac1n k(x, x_n) \end{bmatrix} \begin{bmatrix} y_1 \\ \vdots \\ y_n \end{bmatrix}.$$

Stacking this up for the different $x_i$ gives $\hat f = S y$. Thus each row of $S$, the effective kernel, is the kernel evaluation with each of the input points, divided by $n$.

If you change the bandwidth, the effective kernel changes according to the new bandwidth.

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