Stochastic kernel

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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

where y_i is the observed series, y is its mean, λ is the bandwith, and K is the kernel function.

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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 / 26(1 − t²)² for − 1 < t < 1.
• The Epanechnikov kernel is K = 3 / 4(1 − t²) for − 1 < t < 1.

Nonparametric Econometrics

by Adrian Pagan and Aman Ullah

there are many other books about this topic.

http://ideas.repec.org/a/cup/etheor/v11y1995i3p560-96.html

http://www.uc.edu/sashtml/insight/chap39/sect44.htm