Multivariate Locally Adaptive Density Estimation

SUMMARY: Multivariate versions of variable bandwidth kernel density estimators can be used to combat the eeects of the curse of dimensionality. They are also more exible than the xed bandwidth estimator to model complex (multimodal) densities. In this work, two variable bandwidth estimators are discussed: the balloon estimator which varies the smoothing matrix with each estimation point and the sample point estimator which uses a diierent smoothing matrix for each data point. Binning is used to compute the mean integrated squared error (MISE) for the sample point estimator. This allows a direct comparison between the xed, sample point, and the balloon estimator. Both adaptive estimators show considerable improvement in terms of MISE over the xed bandwidth estimator, and the sample point estimator beats the balloon estimator in two dimensions with moderate sample sizes. Parameterization of the smoothing matricies is also discussed and it is shown that while a more restrictive smoothing matrix can lead to improved estimates in practice, a more general smoothing matrix can lead to serious problems.