Kernel regression

The Kernel regression is a non-parametrical techniques in statistics to estimate the conditional expectation of random variable.

In any non-parametric regression, the conditional expectation of a variable $Y$ relative to a variable $X$ may be written:

$\operatorname {E} (Y|X)=m(X)$

where $m$ is a non-parametric function.

Nadarya (1964) and Watson (1964) proposed to estimate $m$ as a locally weighted average, using a kernel as a weighting function. The Nadarya-Watson estimator is:

${\widehat {m}}_{h}={\frac {n^{-1}\sum _{i=1}^{n}K_{h}(x-X_{i})Y_{i}}{n^{-1}\sum _{i=1}^{n}K_{h}(x-X_{i})}}$

where $K$ is a kernel with a bandwith $h$.