L1 Properties of the Nadaraya Quantile Estimator

Let X be a real random variable having f as density function. Let F be its cumulative distribution function and Q its quantile function. For h > 0, let F h and Q h denote respectively the Nadaraya kernel estimator of F and Q . In the first part of this paper the almost sure convergence of the conventional L 1 distance between Q h and Q is established. In the second part, the L 1 right inversion distance is introduced. The representation of this L 1 right inversion distance in terms of F h and F is given. This representation allows us to suggest ways to choose a global bandwidth for the estimator Q h .