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

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

where $y_{i}$ is the observed series, $\lambda$ 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 $-1<t<1$.

External links

• Example using stochastic kernel for regression (Kardi Teknomo's tutorial)
• Conditional Stochastic Kernel Estimation by Nonparametric Methods (Laurini, Márcio P. & Valls Pereira, Pedro L.)