Minimizing L1 distance in nonparametric density estimation

We construct a simple algorithm, based on Newton's method, which permits asymptotic minimization of L1 distance for nonparametric density estimators. The technique is applicable to multivariate kernel estimators, multivariate histogram estimators, and smoothed histogram estimators such as frequency polygons. It has an "adaptive" or "data-driven" version. We show theoretically that both theoretical and adaptive forms of the algorithm do indeed minimize asymptotic L1 distance. Then we apply the algorithm to derive concise formulae for asymptotically optimal smoothing parameters. We also give numerical examples of applications of the adaptive algorithm.