Consistency of asymmetric kernel density estimators and smoothed histograms with application to income data

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 microdata set are finally provided.