Boundary Performance of the Beta Kernel Intensity Estimators for Inhomogeneous Poisson Process

Abstract

We consider the non-parametric estimation of the bivariate intensity function of non-homogeneous Poisson processes in a bounded data. To this end we use the class of kernel estimators with asymmetric beta kernel functions. The beta kernels are non-negative.
They change their shape depending on the position on the semi-axis and possess good boundary properties for a wide class of intensity. The theoretical asymptotic properties of the bivariate intensity function like bias and variance are derived. We obtain the optimal bandwidth selection for both estimates as a minimum of the mean integrated squared error (MISE). Numerical studies indicate that the performance of our approach is better, comparing with other bandwidth selection techniques using  bias, variance and integrated squared error as criterion. Some applications are made on real datasets.