Gram-Charlier A Series Based Extended Rule-of-Thumb for Bandwidth Selection in Univariate Kernel Density Estimation

Abstract

Bandwidth parameter estimation in univariate Kernel Density Estimation has traditionally two approaches. Rule(s)-of-Thumb (ROT) achieve ‘quick and dirty’ estimations with some specific assumption for an unknown density. More accurate solve-the-equation-plug-in (STEPI) rules have almost no direct assumption for the unknown density but demand high computation. This article derives a balancing third approach. Extending an assumption of Gaussianity for the unknown density to be estimated in \textit{normal reference} ROT (NRROT) to near Gaussianity, and then expressing the density using Gram-Charlier A (GCA) series to minimize the asymptotic mean integrated square error, it derives GCA series based Extended ROT (GCAExROT). The performance analysis using the simulated and the real datasets suggests to replace NRROT by a modified GCAExROT rule achieving a balancing performance by accuracy nearer to STEPI rules at computation nearer to NRROT, specifically at small samples.