## Quick Reference

A method for the estimation of probability density functions. Suppose *X* is a continuous random variable with unknown probability density function f. A random sample of observations of *X* is taken. If the sample values are denoted by *x*_{1}, *x*_{2},…, *x** _{n}*, the estimate of f is given by where K is a kernel function and the constant

*A*is chosen so that The observation

*x*

*may be regarded as being spread out between*

_{j}*x*

*−*

_{j}*a*and

*x*

*+*

_{j}*b*(usually with

*a*=

*b*). The result is that the naive estimate of f as being a function capable of taking values only at

*x*

_{1},

*x*

_{2},…,

*x*

*, is replaced by a continuous function having peaks where the data are densest. Examples of kernel functions are the Gaussian kernel, and the Epanechikov kernel,, The constant*

_{n}*h*is the window width or bandwidth. The choice of

*h*is critical: small values may retain the spikes of the naive estimate, and large values may oversmooth so that important aspects of f are lost.,

**Kernel method.** In this case a sample of twenty observations have been generated randomly from a chi-squared distribution with twenty degrees of freedom and a Gaussian kernel with *h*=3 has been used to generate the kernel density estimate.

*Subjects:*
Probability and Statistics.

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