A weighted univariate kernel density estimate involves a variable X and a weight variable W. Let , denote a sample of X and W of size n. The weighted kernel density estimate of , the density of X, is as follows:

where h is the bandwidth and

is the standard normal density rescaled by the bandwidth. If and , then the optimal bandwidth is

This optimal value is unknown, and so approximations methods are required. For a derivation and discussion of these results, see Silverman (1986, ChapterĀ 3) and Jones, Marron, and Sheather (1996).

For the bivariate case, let be a bivariate random element taking values in with joint density function

and let , be a sample of size n drawn from this distribution. The kernel density estimate of based on this sample is

where , and are the bandwidths, and is the rescaled normal density

where is the standard normal density function

Under mild regularity assumptions about , the mean integrated squared error (MISE) of is

as , and .

Now set

which is the asymptotic mean integrated squared error (AMISE). For fixed n, this has a minimum at defined as

and

These are the optimal asymptotic bandwidths in the sense that they minimize MISE. However, as in the univariate case, these expressions contain the second derivatives of the unknown density being estimated, and so approximations are required. See Wand and Jones (1993) for further details.