Kernel regression

Remarks

The weights $K((x - X_i)/h) / \sum_j K((x - X_j)/h)$ are nonnegative and sum to $1$, so $\hat m_h(x)$ is a weighted average of the $Y_i$. Dividing the numerator and the denominator by $nh$ shows the denominator is the kernel density estimate of the law of $X$ at $x$. Example: with the box kernel $K = \frac{1}{2} \mathbf{1}_{[-1, 1]}$, $\hat m_h(x)$ is the plain average of the $Y_i$ with $\abs{X_i - x} \le h$. Non-example: the numerator alone, divided by $nh$, estimates $m(x) f_X(x)$ rather than $m(x)$, where $f_X$ is the density of $X$. Local polynomial regression generalizes this estimator; the Nadaraya–Watson estimator is the local constant case.