Stochastic kernel estimation

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

Examples

• The uniform kernel is $1/2$ for $-1<t<1$.
• The triangular kernel is $1-|t|$ for $-1<t<1$.
• The quartic kernel is $15/26(1-t^{2})^{2}$ for $-1<t<1$.
• The Epanechnikov kernel is $3/4(1-t^{2})$ for $-1<t<1$.

Often, the data is fitted to such a kernel by setting a window width h, considering only $x_{i}$'s in $x\pm h/2$ and setting $t_{i}=(x_{i}-x)/h$.