Finite Sample Error Bound for Parzen Windows

Abstract

Parzen Windows as a nonparametric method has been applied to a variety of density estimation as well as clas-sification problems. Similar to nearest neighbor meth-ods, Parzen Windows does not involve learning. While it converges to true but unknown probability densities in the asymptotic limit, there is a lack of theoretical anal-ysis on its performance with finite samples. In this pa-per we establish a finite sample error bound for Parzen Windows. We first show that Parzen Windows is an ap-proximation to regularized least squares (RLS) methods that have been well studied in statistical learning theory. We then derive the finite sample error bound for Parzen Windows, and discuss the properties of the error bound and its relationship to the error bound for RLS. This analysis provides interesting insight to Parzen Windows as well as the nearest neighbor method from the point of view of learning theory. Finally, we provide em-pirical results on the performance of Parzen Windows and other methods such as nearest neighbors, RLS and SVMs on a number of real data sets. These results cor-roborate well our theoretical analysis.