Data Dependent Asymmetric Kernels for Estimating the Density Function

Abstract

In the nonparametric setup, smooth density estimators are obtained by using a kernel function and it is used to account for contribution of each observation to the estimator. Generally, these kernel functions are symmetric about zero. We propose an estimator by using data dependent asymmetric kernels being generated by an unimodal symmetric kernel. To account for the contribution of the ith order statistic to the estimator, a suitable asymmetric kernel is used. Asymptotic Bias, Mean Square Error (MSE) and Mean Integrated Square Error (MISE) of the estimator are obtained with higher accuracies by using some properties of the densities of order statistics. The convergence rates depend on both n, the sample size and hn, the band width. Expressions for the optimum band width minimizing the asymptotic MISE and the rate of convergence of the optimum MISE are obtained. Simulation study and data analysis indicate that the proposed estimator not only reduces bias outside the support significantly but also it is closer to the true density function as compared to that of the usual estimator based on a common symmetric kernel. Scope for the multivariate generalization of the method is given. Some methods of generating multivariate skew symmetric kernels are described. As an illustration one of them has been used in the simulation study and estimation of density in bivariate case.