We consider asymmetric kernel density estimators and smoothed histograms when the unknown probability density function f is defined on @0,ϩ`!+ Uniform weak consistency on each compact set in @0,ϩ`! 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 ϭ 0 when the density is unbounded at x ϭ 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+ INTRODUCTIONThe most popular nonparametric estimator of an unknown probability density function f is the standard kernel estimator+ Its consistency is well documented when the support of the underlying density is unbounded+ In the case of a bounded support we know that there exists a boundary bias~see, e+g+, the estimation of Figure 3 in Section 5!+ This problem is due to the use of a fixedWe 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

In this paper we investigate a new approach of estimating a regression function based on copulas.The main idea behind this approach is to write the regression function in terms of a copula and marginal distributions. Once the copula and the marginal distributions are estimated we use the plug-in method to construct the new estimator. Because various methods are available in the literature for estimating both a copula and a distribution, this idea provides a rich and flexible alternative to many existing regression estimators. We provide some asymptotic results related to this copula-based regression modeling when the copula is estimated via profile likelihood and the marginals are estimated nonparametrically. We also study the finite sample performance of the estimator and illustrate its usefulness by analyzing data from air pollution studies. * H. Noh acknowledges financial support from IAP research network P6/03 of the Belgian Government (Belgian Science Policy). † A. El Ghouch acknowledges financial support from IAP research network P6/03 of the Belgian Government (Belgian Science Policy), and from the contract 'Projet d'Actions de Recherche Concertées' (ARC) 11/16-039 of the 'Communauté française de Belgique', granted by the 'Académie universitaire Louvain'.

This paper proposes a new nonparametric test for conditional independence, which is based on the comparison of Bernstein copula densities using the Hellinger distance. The test is easy to implement because it does not involve a weighting function in the test statistic, and it can be applied in general settings since there is no restriction on the dimension of the data. In fact, to apply the test, only a bandwidth is needed for the nonparametric copula. We prove that the test statistic is asymptotically pivotal under the null hypothesis, establish local power properties, and motivate the validity of the bootstrap technique that we use in finite sample settings. A simulation study illustrates the good size and power properties of the test. We illustrate the empirical relevance of our test by focusing on Granger causality using financial time series data to test for nonlinear leverage versus volatility feedback effects and to test for causality between stock returns and trading volume. In a third application, we investigate Granger causality between macroeconomic variables. JEL Classification: C12; C14; C15; C19; G1; G12; E3; E4; E52.

We propose a new nonparametric estimator for the density function of multivariate bounded data. As frequently observed in practice, the variables may be partially bounded (e.g., nonnegative) or completely bounded (e.g., in the unit interval). In addition, the variables may have a point mass. We reduce the conditions on the underlying density to a minimum by proposing a nonparametric approach. By using a gamma, a beta, or a local linear kernel (also called boundary kernels), in a product kernel, the suggested estimator becomes simple in implementation and robust to the well known boundary bias problem. We investigate the mean integrated squared error properties, including the rate of convergence, uniform strong consistency and asymptotic normality. We establish consistency of the least squares cross-validation method to select optimal bandwidth parameters. A detailed simulation study investigates the performance of the estimators. Applications using lottery and corporate finance data are provided.

Nonparametric estimation for an unknown probability density function f with a known compact support [0, 1] not necessarily bounded at x = 0 is considered. For such class of density functions, we consider the Bernstein estimator. The uniform weak consistency and the uniform strong consistency on each compact I in (0, 1) are established for the Bernstein estimator. We prove also the almost sure convergence to infinity at x = 0 of the Bernstein estimator when the density function f is unbounded at x = 0. To select the optimal bandwidth parameter of the Bernstein estimator, the least squares cross-validation and the likelihood cross-validation methods are developed.