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

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 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).

On Hybridizations of Fourth Order Kernel of the Beta Polynomial Family.

The usual second order nonparametric kernel estimators are of wide uses in data analysis and visualization but constrained with slow convergence rate. Higher order kernels provide a faster convergence rates and are known to be bias reducing kernels. In this paper, we propose a hybrid of the fourth order kernel which is a merger of two successive fourth order kernels and the statistical properties of these hybrid kernels were study. The results of our simulation reveals that the proposed higher order hybrid kernels outperformed their corresponding parent’s kernel functions using the asymptotic mean integrated squared error