Effects of Non-Gaussian Kernels on Multimodality Investigations

• Activity Number: 207
• Type: Topic Contributed
• Date/Time: Tuesday, August 13, 2002 : 10:30 AM to 12:20 PM
• Sponsor: IMS
• Author(s): Michael Minnotte*+ and Peter Hall and Chunming Zhang
• Affiliation(s): Utah State University and Australian National University and University of Wisconsin, Madison
• Keywords: Bump Hunting ; Critical Bandwidth ; Density Estimation ; Kernel Choice ; Modal Monotonicity ; Mode Testing

Abstract

It is well known that the number of modes of a kernel density estimator is monotone nonincreasing in the bandwidth if the kernel is a normal density. There is numerical evidence of nonmonotonicity in the case of some non-Gaussian kernels, but little additional information is available. We provide theoretical and numerical descriptions of the extent to which the number of modes is a non-monotone function of bandwidth in the case of general compactly supported densities. Our results address popular kernels used in practice--for example, the Epanechnikov, biweight, and triweight kernels, and show that in such cases non-monotonicity is present with strictly positive probability for all sample sizes greater than two. Nevertheless, in spite of the prevalence of lack of monotonicity revealed by these results, it is shown that the notion of a critical bandwidth (the smallest bandwidth above which the number of modes is guaranteed to be monotone) is still well-defined. Implications for bump hunting using such kernels will be discussed.