Multivariate kernel density estimation: Difference between revisions

Revision as of 07:54, 30 March 2016

Revision as of 07:57, 30 March 2016

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

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

It can be shown that any reasonable bandwidth selector '''H''' has '''H''' = ''O''(''n''^{−2/(''d''+4)})'' where the [[big O notation]] is applied elementwise. Substituting this into the MISE formula yields that the optimal MISE is ''O''(''n''^{−4/(''d''+4)}).''<ref name="WJ1995"/>{{rp|99–100}} Thus as ''n'' → ∞'', the MISE → 0, i.e. the kernel density estimate [[convergence in mean|converges in mean square]] and thus also in probability to the true density ''f''. These modes of convergence are confirmation of the statement in the motivation section that kernel methods lead to reasonable density estimators. An ideal optimal bandwidth selector is

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

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Silverman's rule of thumb suggests using <math>\sqrt{\mathbf{H}_{ii}} = \left(\frac{4}{d+2}\right)^{\frac{1}{d+4}} n^{\frac{-1}{d+4}} \sigma_i</math> where <math>\sigma_i</math> is the standard deviation of the ith variable and <math>\mathbf{H}_{ij} = 0, i\neq j</math>. Scott's rule is <math>\sqrt{\mathbf{H}_{ii}} = n^{\frac{-1}{d+4}} \sigma_i</math>.

==Asymptotic analysis==