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:<math>\operatorname{vec} (\hat{\mathbf{H}} - \mathbf{H}_{\operatorname{AMISE}}) = O(n^{-2\alpha}) \operatorname{vec} \mathbf{H}_{\operatorname{AMISE}}.</math>
It has been established that the plug-in and smoothed cross validation selectors (given a single pilot bandwidth '''G''') both converge at a relative rate of ''O<sub>p</sub>''(''n''<sup>−2/(''d''+6)</sup>) <ref name="DH2005" /><ref>{{Cite journal| doi=10.1016/j.jmva.2004.04.004 | author1=Duong, T. | author2=Hazelton, M.L. | title=Convergence rates for unconstrained bandwidth matrix selectors in multivariate kernel density estimation | journal=Journal of Multivariate Analysis | year=2005 | volume=93 | issue=2 | pages=417–433| doi-access=free }}</ref> i.e., both these data-based selectors are consistent estimators.
==Density estimation with a full bandwidth matrix==
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