We consider asymmetric kernel density estimators and smoothed histograms when the unknown probability density function f is defined on [0,+infinity). Uniform weak consistency on each compact set in [0,+infinity) is proved for these estimators when f is continuous on its support. Weak convergence in L_1 is also established. We further prove that the asymmetric kernel density estimator and the smoothed histogram converge in probability to infinity at x=0 when the density is unbounded at x=0. Monte Carlo results and an empirical study of the shape of a highly skewed income distribution based on a large micro-data set are finally provided.

Keywords : Asymmetric kernel, smoothed histogram, density estimation, weak convergence, L_1 consistency, unbounded density, boundary bias, income distribution, inequality measurement.

JEL : C13,C14.
MSC 2000 : 62G07, 62G08.