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The article derives a novel Gram-Charlier A (GCA) Series based ExtendedRule-of-Thumb (ExROT) for bandwidth selection in Kernel Density Estimation(KDE). There are existing various bandwidth selection rules achievingminimization of the Asymptotic Mean Integrated Square Error (AMISE) between theestimated probability density function (PDF) and the actual PDF. The rulesdiffer in a way to estimate the integration of the squared second orderderivative 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 bandwidthselection rules can be derived, if the unknown PDF is approximated through aninfinite series expansion based on a more generalized density assumption. As ademonstration and verification to this concept, the ExROT derived in thearticle uses an extended assumption that the density being estimated is nearGaussian. This helps use of the GCA expansion as an approximation to theunknown near Gaussian PDF. The ExROT for univariate KDE is extended to that formultivariate KDE. The required multivariate AMISE criteria is re-derived usingelementary calculus of several variables, instead of Tensor calculus. Thederivation uses the Kronecker product and the vector differential operator toachieve the AMISE expression in vector notations. There is also derived ExROTfor kernel based density derivative estimator.

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