Smooth copula-based estimation of the conditional density function with a single covariate

Some recent papers deal with smooth nonparametric estimators for copula functions and copula derivatives. These papers contain results on copula-based Bernstein estimators for conditional distribution functions and related functionals such as regression and quantile functions. The focus in the present paper is on new copula-based smooth Bernstein estimators for the conditional density. Our approach avoids going through separate density estimation of numerator and denominator. Our estimator is defined as a smoother of the copula-based Bernstein estimator of the conditional distribution function. We establish asymptotic properties of bias and variance and discuss the asymptotic mean squared error in terms of the smoothing parameters. We also obtain the asymptotic normality of the new estimator. In a simulation study we show the good performance of the new estimator in comparison with other estimators proposed in the literature.