Analyzing the Effect of Adaptive Bandwidth Selection in Nonparametric Density Estimation and Smoothing

Posted: Sep 11, 2018

• Daniel Wood
• David Gray
• Dylan Turner

Abstract

Nonparametric density estimation represents a fundamental tool in statistical analysis and machine learning, providing flexible approaches for modeling data distributions without strong parametric assumptions. The kernel density estimator, introduced by Rosenblatt and Parzen, has become one of the most widely used methods in this domain. However, the performance of kernel density estimation critically depends on the selection of an appropriate bandwidth parameter, which controls the degree of smoothing applied to the data. Traditional approaches to bandwidth selection, including rules-of-thumb and cross-validation methods, typically employ a single global bandwidth that applies uniformly across the entire data space. This uniform treatment often proves suboptimal for real-world datasets that exhibit heterogeneous characteristics, such as varying local densities, multiple modes, or complex spatial structures. The limitations of global bandwidth selection have motivated research into adaptive methods that allow bandwidth parameters to vary according to local data characteristics. While several adaptive approaches have been proposed in the literature, including Abramson's square root law and sample point smoothing methods, these techniques often face challenges related to computational complexity, sensitivity to initial conditions, and theoretical justification. Moreover, existing adaptive methods frequently struggle to balance the competing demands of local adaptation and global coherence, sometimes producing density estimates that appear artificially fragmented or that fail to capture important global patterns. This paper addresses these challenges by introducing a novel adaptive bandwidth selection framework that combines principles from computational geometry and information theory. Our approach leverages Voronoi tessellation to partition the data space according to local density characteristics, then employs entropy-based optimization to determine appropriate bandwidth parameters within each partition. This hybrid methodology enables sophisticated local adaptation while maintaining computational tractability and theoretical soundness.