Multivariate kernel density estimation: Difference between revisions

Revision as of 09:50, 19 September 2010

Revision as of 10:33, 20 September 2010

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[[Kernel density estimation]] is a technique for [[nonparametric]] technique for [[density estimation]] i.e., estimation of [[probability density function]]s, which is one of the fundamental questions in [[statistics]].

It can be viewed as a generalisation of [[histogram]] density estimation with improved statistical properties.

Apart from histograms, other types of density estimators include [[parametric statistics | parametric]], [[spline interpolation |spline]], [[wavelet]] and [[Fourier series]].

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where <math>\hat{\bold{\Psi}}_4 (\bold{G}) = n^{-2} \sum_{i=1}^n

\sum_{j=1}^n [(\operatorname{vec} \, \operatorname{D}^2) (\operatorname{vec}^T \operatorname{D}^2)] K_\bold{G} (\bold{X}_i - \bold{X}_j)</math>. Thus <math>\hat{\bold{H}}_{\operatorname{PI}} = \operatorname{argmin}_{\bold{H} \in F} \, \operatorname{PI} (\bold{H})</math> is the plug-in selector.<ref>{{Cite journal| author1=Wand, M.P. | author2=Jones, M.C. | title=Multivariate plug-in bandwidth selection | journal=Computational Statistics | year=1994 | volume=9 | pages=97–177}}</ref><ref>{{Cite journal| doi=10.1080/10485250306039 | author1=Duong, T. | author2=Hazelton, M.L. | title=Plug-in bandwidth matrices for bivariate kernel density estimation | journal=Journal of Nonparametric Statistics | year=2003 | volume=15 | pages=17–30}}</ref>. These references also contain algorithms on optimal estimation of the pilot bandwidth matrix <strong>G</strong> and establish that <math>\hat{\bold{H}}_{\operatorname{PI}}</math> [[convergence in probability|converges in probability]] to <math>\bold{H}_{\operatorname{AMISE}}</math>.

===Smoothed cross validation===

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+ K_{2\bold{G}}) (\bold{X}_i - \bold{X}_j)</math>

Thus <math>\hat{\bold{H}}_{\operatorname{SCV}} = \operatorname{argmin}_{\bold{H} \in F} \, \operatorname{SCV} (\bold{H})</math> is the SCV selector.<ref>{{Cite journal| doi=10.1007/BF01205233 | author1=Hall, P. | author2=Marron, J. | author3=Park, B. | title=Smoothed cross-validation | journal=Probability Theory and Related Fields | year=1992 | volume=92 | pages=1–20}}</ref><ref>{{Cite journal| doi=10.1111/j.1467-9469.2005.00445.x | author1=Duong, T. | author2=Hazelton, M.L. | title=Cross validation bandwidth matrices for multivariate kernel density estimation | journal=Scandinavian Journal of Statistics | year=2005 | volume=32 | pages=485–506}}</ref>.

These references also contain algorithms on optimal estimation of the pilot bandwidth matrix <strong>G</strong> and establish that <math>\hat{\bold{H}}_{\operatorname{SCV}}</math> converges in probability to <math>\bold{H}_{\operatorname{AMISE}}</math>.