The Gram-Charlier A Series based Extended Rule-of-Thumb for Bandwidth Selection in Univariate and Multivariate Kernel Density Estimations

The article derives a novel Gram-Charlier A (GCA) Series based Extended Rule-of-Thumb (ExROT) for bandwidth selection in Kernel Density Estimation (KDE). There are existing various bandwidth selection rules achieving minimization of the Asymptotic Mean Integrated Square Error (AMISE) between the estimated probability density function (PDF) and the actual PDF. The rules differ in a way to estimate the integration of the squared second order derivative of an unknown PDF \((f(\cdot))\), identified as the roughness \(R(f''(\cdot))\). The simplest Rule-of-Thumb (ROT) estimates \(R(f''(\cdot))\) with an assumption that the density being estimated is Gaussian. Intuitively, better estimation of \(R(f''(\cdot))\) and consequently better bandwidth selection rules can be derived, if the unknown PDF is approximated through an infinite series expansion based on a more generalized density assumption. As a demonstration and verification to this concept, the ExROT derived in the article uses an extended assumption that the density being estimated is near Gaussian. This helps use of the GCA expansion as an approximation to the unknown near Gaussian PDF. The ExROT for univariate KDE is extended to that for multivariate KDE. The required multivariate AMISE criteria is re-derived using elementary calculus of several variables, instead of Tensor calculus. The derivation uses the Kronecker product and the vector differential operator to achieve the AMISE expression in vector notations. There is also derived ExROT for kernel based density derivative estimator.