Abstract
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 Fh and Qh denote respectively the Nadaraya kernel estimator of F and Q. In the first part of this paper the almost sure convergence of the conventional L1 distance between Qh and Q is established. In the second part, the L1 right inversion distance is introduced. The representation of this L1 right inversion distance in terms of Fh and F is given. This representation allows us to suggest ways to choose a global bandwidth for the estimator Qh.
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