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<ul>
<li><math>R(K) = \int K(\bold{x})^2 \, d\bold{x}</math>. For the normal kernel <math>K</math>, <math>R(K) = (4 \pi)^{-d/2}</math>
<li><math>\int \bold{x} \bold{x}^T K(\bold{x})^2 \, d\bold{x}</math> = m_2(K) \bold{I}_d</math>,
with <math>\bold{I}_d</math> is the <em>d x d</em> [[identity matrix]]
<li><math>\operatorname{D}^2 f</math> is the <em>d x d</em> Hessian matrix of second order partial derivatives of <math>f</math>
</ul>
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\int \operatorname{tr}^2 (\bold{H} \widehat{\operatorname{D}^2 f} (\bold{x})) \, d\bold{x}</math>
and <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>
=== Smoothed cross validation ===
Smoothed cross validation is a subset of a larger class of [[cross validation]] techniques.
<ref name="CD2010">{{cite journal|doi=10.1007/s11749-009-0168-4 | author1=Chacón, J.E | author2=Duong, T. | title=Multivariate plug-in bandwidth selection with unconstrained pilot bandwidth matrices | journal=[[Test]] | year=2010 | volume=19 | pages=375-398}}</ref>.
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