A kernel is a weighting function used in non-parametric estimation techniques. Kernels are used in kernel density estimation to estimate random variables' density functions, or in kernel regression to estimate the conditional expectation of a random variable. Kernels are also used in time-series, in the use of the periodogram to estimate the spectral density. An additional use is in the estimation of a time-varying intensity for a point process.
Commonly, kernel widths must also be specified when running a non-parametric estimation.
The first requirement ensures that the method of kernel density estimation results in a probability density function. The second requirement ensures that the average of the corresponding distribution is equal to that of the sample used.
If K is a kernel, then so is the function K* defined by K*(u) = λ−1K(λ−1u), where λ > 0. This can be used to select a scale that is appropriate for the data.
Below, the notation denotes the value 1 when p holds, and 0 when p is false.