Stochastic kernel estimation

A stochastic kernel is the transition function of a (usually discrete-time) stochastic process. Often, it is assumed to be iid, thus a probability density function.

Formally a density can be

$f_{\lambda }(y_{i})={\frac {1}{I\lambda }}\sum _{i=1}^{I}K\left({\frac {y-y_{i}}{\lambda }}\right),$

where $y_{i}$ is the observed series, y is its mean, $\lambda$ is the bandwidth, and K is the kernel function.

Examples

• The uniform kernel is $K=1/2$ for $-1<t<1$.
• The triangular kernel is $K=1-|t|$ for $-1<t<1$.
• The quartic kernel is $K=15/16(1-t^{2})^{2}$ for $-1<t<1$.
• The Epanechnikov kernel is $K=3/4(1-t^{2})$ for $-1<t<1$.