On the estimation of monotone uniform approximations

This paper studies the estimation of L[infinity]-best monotone approximations to a - known or unknown - function H. When H is known, natural empirical estimates are pointwise consistent and also uniform consistent under an extra continuity condition. When H is unknown, a preliminary uniform estimation of H is shown to be necessary so that the method can be applied to this estimate instead of H. However, some points of application do not allow this possibility and therefore a more general procedure of monotone approximation through appropriate sample subsets is investigated. It is shown to be pointwise consistent, and a uniform consistency through these subsets is also assured. The method is applied in Nonparametric Regression.L[infinity]-best monotone approximations L[infinity]-Dip Uniform consistency Regression function Kernel regression estimate