Multivariate kernel density estimation: Difference between revisions

Revision as of 12:20, 15 September 2010

Revision as of 13:19, 15 September 2010

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where

<ul>

<li><math>\bold{x} = (x_1, x_2, \dots, x_d)^T</math>, <math>\bold{X}_i = (X_{i1}, X_{i2}, \dots, X_{id})^T, i=1, 2, \dots, n.</math> are <em>d</em>-vectors

<li><em>K</em> is the kernel function which is a symmetric density function, with <math>K_\bold{H}(\bold{x}) = |\bold{H}|^{-1/2} K(\bold{H}^{-1/2} \bold{x})</math>

<li><strong>H</strong> is the bandwidth (or smoothing) matrix which is a symmetric, [[positive definite matrix|positive definite]] <em>d x d</em> matrix.

</ul>

The choice of the kernel function <em>K</em> is not crucial to the accuracy of kernel density estimators, whereas the choice of the bandwidth matrix <strong>H</strong> is the single most important factor affecting its accuracy.<ref>{{cite book | author1=Wand, M.P | author2=Jones, M.C. | title=Kernel Smoothing | publisher=Chapman Hall | ___location=London | date=1995 | page=36-39 | isbn = 0412552701}}</ref>

== Optimal bandwidth matrix selection ==

== References ==