Kernel Regression

Definition

Kernel regression is an extension of Kernel Density Estimation to estimate the conditional expectation of a random variable $Y$ given $X$. The kernel regression estimator is defined as ${\hat{f}}_{\lambda }(x)=\frac{1}{n}\sum _{i=1}^n{K}_{\lambda }(x-X_i)Y_i=\frac{1}{n\lambda }\sum _{i=1}^{n}K(\frac{x-X_i}{\lambda })Y_i$ where $K\text{s.t.}{\int }_{-\infty }^{\infty }K(t)dt=1,{\int }_{-\infty }^{\infty }{K}^2(t)dt<\infty$ is the kernel, $K_{\lambda }(x):=\frac{1}{h}K(\frac{x}{h})$ is the scaled kernel, and $\lambda$ is a smoothing parameter