Smooth conditional distribution estimators using Bernstein polynomials

In a variety of statistical problems, estimation of the conditional distribution function remains a challenge. To this end, a two-stage Bernstein estimator for conditional distribution functions is introduced. The method consists in smoothing a first-stage Nadaraya-Watson or local linear estimator by constructing its Bernstein polynomial. Some asymptotic properties of the proposed estimator are derived, such as its asymptotic bias, variance and mean squared error. The asymptotic normality of the estimator is also established under appropriate conditions of regularity. Lastly, the performance of the proposed estimator is briefly studied through a few examples.