Multivariate kernel density estimation: Difference between revisions

Revision as of 13:50, 15 September 2010

Revision as of 14:13, 15 September 2010

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This is in general does not possess a closed form expression, so it is usual to use its asymptotic approximation (AMISE) as a proxy

<math>\operatorname{AMISE} (\bold{H}) = n^{-1} |\bold{H}|^{-1/2} R(K) + \tfrac{1}{4} m_2(K)^2

(\operatorname{vec}^T \bold{H}) \bold{R} (\operatorname{vec} \, \operatorname{D}^2 f) (\operatorname{vec} \bold{H}) </math>

where

<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>\operatorname{D}^2 f</math> is the <em>d x d</em> Hessian matrix of second order partial derivatives of <math>f</math>

<li><math>\bold{R}(\operatorname{D}^2 f) = \int (\operatorname{vec} \, \operatorname{D}^2 f (\bold{x})) (\operatorname{vec} \, \operatorname{D}^2 f (\bold{x}))^T \, d\bold{x}</math>

<li>vec is the vector operator which stacks the columns of matrix into a single vector e.g.

<math>\operatorname{vec} \begin{bmatrix} a & c & e\\ b & d & f\end{bmatrix} = \begin{bmatrix} a & b & c & d & e & f\end{bmatrix}^T</math>

</ul>

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

where [[Big O notation | o, O]] indicate the usual small and [[big O notation]]. Heuristically this statement implies that the AMISE is a 'good' approximation of the MISE as the sample size <em>n → ∞<em>.

The many different varieties of bandwidth selectors arise from the different estimators of the MISE or AMISE. We concentrate on two classes of selectors which have been shown to be the most widely applicable in practise: smoothed cross validation and plug-in selectors.

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=== Smoothed cross validation ===

== References ==