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 x1, x2,…, xn, the estimate of f is given by where K is a kernel function and the constant A is chosen so that The observation xj may be regarded as being spread out between xj − a and xj+b (usually with a=b). The result is that the naive estimate of f as being a function capable of taking values only at x1, x2,…, xn, 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 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.