Generalization Of Asymmetric Kernel Density Estimation Under Multidimensional Distribution And Grouped Data

Kernel density estimation is a data-driven approach of estimating an unknown prob-ability density based on a finite data sample.The standard kernel density estimator uses symmetric kernel function. The method is intuitively appealing and easy to implement, but it suffers from boundary bias, lack of local adaptivity, and the tendency to flatten out peaks and valleys of the true density. Asymmetric kernel density estimators are pro-posed to address these problems. The shape of these kernel functions varies with the position of the data points, changing the degree of smoothing automatically, and thus are free of boundary bias, and achieve the optimal rate of convergence for the mean inte-grated squared error within the class of non-negative kernel density estimators. We first establish the strong convergence of asymmetric kernel density estimators in L1. Then we generalise the asymmetric kernel density estimation to the multi-dimensional cases by employing the ideas of copulas. We propose a semi-parametric estimator which combines RIG kernel for the estimation of the marginal densities with parametric copulas to model the dependence within the variables. This semi-parametric estimator is free of boundary bias and and avoids the problem of curse of dimensionality. Uniform strong consistency and asymmetric normality are established and mean square error properties are derived. Finally, we consider asymmetric kernel estimates based on grouped data. We propose an iterated scheme for constructing such an estimator and apply an iterated smoothed bootstrap approach for bandwidth selection. We compare our approach with competing methods in estimating actuarial loss models using both simulations and data studies. The simulation results show that with this new method, the estimated density from grouped data matches the true density more closely than with competing methods.