A Novel Nonparametric Two-Sample Hypothesis Test Using Geometric Formulations

• Activity Number: 296
• Type: Contributed
• Date/Time: Tuesday, August 5, 2014 : 8:30 AM to 10:20 AM
• Sponsor: Section on Nonparametric Statistics
• Author(s): Zhengwu Zhang*+ and Anuj Srivastava and Eric Klassen
• Companies: FSU and Florida State University and Florida State University
• Keywords: kernel estimation ; heat equation ; bandwidth-invariant ; hypothesis test

Abstract

The estimation and comparison of the probability densities are fundamental problems in statistics. The classical non-parametric method of estimating probability densities using sample data is based on kernel density estimator. However, in kernel density estimation, e.g. when using the Gaussian isotropic kernel, the choice of the bandwidth is crucial and can greatly affect the results. Thus, a comparison of estimated densities will also be affected by the choice of bandwidth. In this paper we describe a framework for metric-based comparisons of estimated densities, estimated using the Gaussian kernel, in a manner that does not depend on the choice of bandwidth. We establish a group of all possible Gaussian kernels and define its action on the density space using the heat equation to model convolution. Then, we define an orthogonal section of this action to represent and compare densities at the same level of smoothness. We use this metric as a statistic for two-sample hypothesis test and compare with Kolmogorov-Smirnov test. Furthermore, we analyze the use of kernel estimated densities in image classification task and demonstrate the improvement obtained using our approach.