A Key Note on Performance of Smoothing Parameterizations in Kernel Density Estimation

Abstract

The univariate kernel density estimator requires one smoothing parameter while the bivariate and other higher dimensional kernel density estimators demand more than one smoothing parameter depending on the form of smoothing parameterizations used. The smoothing parameters of the higher dimensional kernels are presented in a matrix form called the smoothing matrix. The two forms of parameterizations frequently used in higher dimensional kernel estimators are diagonal or constrained parameterization and full or unconstrained parameterization. While the full parameterization has no restrictions, the diagonal has some form of restrictions. The study investigates the performance of smoothing parameterizations of bivariate kernel estimator using asymptotic mean integrated squared error as error criterion function. The results show that in retention of statistical properties of data and production of smaller values of asymptotic mean integrated squared error as tabulated, the full smoothing parameterization outperforms its diagonal counterpart

Keywords:  Smoothing Matrix; Kernel Estimator, Integrated Variance; Integrated Squared Bias; Asymptotic Mean Integration Squared Error (AMISE)