Consistency of Asymmetric Kernel DensityEstimators and Smoothed Histograms withApplication to Income Data

We consider asymmetric kernel density estimators and smoothed histogramswhen the unknown probability density function f is defined on [0, +∞). Uniform weak consistency on each compact set in [0, +∞) is proved for these estimators when "f" is continuous on its support. Weak convergence in L 1 is alsoestablished. We further prove that the asymmetric kernel density estimatorand the smoothed histogram converge in probability to infinity at x = 0 whenthe density is unbounded at x = 0. Monte Carlo results and an empirical studyof the shape of a highly skewed income distribution based on a large micro-dataset are finally provided.