Multivariate kernel density estimation: Difference between revisions

Revision as of 11:32, 16 September 2010

Revision as of 11:42, 16 September 2010

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<li>vec is the vector operator which stacks the columns of a matrix into a single vector e.g. <math>\operatorname{vec}\begin{bmatrix}a & c \\ b & d\end{bmatrix} = \begin{bmatrix}a & b & c & d\end{bmatrix}^T.</math>

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The quality of the AMISE approximation to the MISE<ref name="WJ1995">/</ref>{{rp|97}} is given by

<math>\operatorname{MISE} (\bold{H}) = \operatorname{AMISE} (\bold{H}) + o(n^{-1} |\bold{H}|^{-1/2}) + O(\operatorname{tr} \, \bold{H}^2)</math>

where <em>o, O</em> indicateindicates the usual small and [[big O notation|small o notation]].<ref name="WJ1995">/</ref>{{rp|97}} Heuristically this statement implies that the AMISE is a 'good' approximation of the MISE as the sample size <em>n → ∞<em>. An ideal optimal bandwidth selector is

<math>\bold{H}_{\operatorname{AMISE}} = \operatorname{argmin}_{\bold{H} \in F} \, \operatorname{AMISE} (\bold{H})</math>