We consider asymmetric kernel density estimators and smoothed
histograms when 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 L1 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.We thank O. Linton and the three
referees for constructive criticism and M.P. Feser and J. Litchfield for
kindly providing the Brazilian data. We are grateful to I. Gijbels, J.M.
Rolin, and I. Van Keilegom for their stimulating remarks and to
participants at the workshop on statistical modeling (UCL 2002), LAMES
(Sao Paulo 2002), L1 Norm conference (Neuchatel 2002), Geneva econometrics
seminar, and KUL econometrics seminar for their comments. Part of this
research was done when the second author was visiting THEMA and IRES. The
first, resp. second, author gratefully acknowledges financial support from
the “Projet d'Actions de Recherche Concertées”
grant 98/03-217, and from the IAP research network grant P5/24 of
the Belgian state, resp. the Swiss National Science Foundation through the
National Centre of Competence in Research: Financial Valuation and Risk
Management (NCCR-FINRISK).