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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 the data-based selector is <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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