Multivariate kernel density estimation: Difference between revisions

Revision as of 20:57, 4 March 2024

Revision as of 23:42, 27 April 2024

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:<math>\operatorname{MSE} \, \hat{f}(\mathbf{x};\mathbf{H}) = \operatorname{Var} \hat{f}(\mathbf{x};\mathbf{H}) + [\operatorname{E} \hat{f}(\mathbf{x};\mathbf{H}) - f(\mathbf{x})]^2</math>

we have that the MSE tends to 0, implying that the kernel density estimator is (mean square) consistent and hence converges in probability to the true density ''f''. The rate of convergence of the MSE to 0 is the necessarily the same as the MISE rate noted previously ''O''(''n''^−4/(d+4)), hence the covergenceconvergence rate of the density estimator to ''f'' is ''O_p''(n^{−2/(''d''+4)}) where ''O_p'' denotes [[Big O in probability notation|order in probability]]. This establishes pointwise convergence. The functional covergenceconvergence is established similarly by considering the behaviour of the MISE, and noting that under sufficient regularity, integration does not affect the convergence rates.

For the data-based bandwidth selectors considered, the target is the AMISE bandwidth matrix. We say that a data-based selector converges to the AMISE selector at relative rate ''O_p''(''n''^−''α''), ''α'' > 0 if