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The `curse of dimensionality' has been interpreted as suggesting that kernel methods have limited applicability in more than several dimensions. In this note, qualitative and quantitative performance measures for multivariate density estimates are examined. Optimal pointwise and global window widths for mean absolute and mean squared errors are compared for multivariate data. One result is that the optimal pointwise absolute and squared error window widths are nearly equal for all dimensions. We also show that sample size requirements predicted by absolute rather than squared error criterion are substantially less. Further reductions are realized by using a coefficient of variation criterion. Finally, an example of a 10-dimensional kernel density estimate is given. It is suggested that the true nature of the curse of dimensionality is as much the lack of full rank as sparseness of the data.
Biometrika © 1991 Biometrika Trust