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On Projection-type Estimators of Multivariate Isotonic Functions


Abdelaati Daouia, Toulouse School of Economics, University of Toulouse, France.


Abstract.  Let M be an isotonic real-valued function on a compact subset of inline image and let inline image be an unconstrained estimator of M. A feasible monotonizing technique is to take the largest (smallest) monotone function that lies below (above) the estimator inline image or any convex combination of these two envelope estimators. When the process inline image is asymptotically equicontinuous for some sequence rn→∞, we show that these projection-type estimators are rn-equivalent in probability to the original unrestricted estimator. Our first motivating application involves a monotone estimator of the conditional distribution function that has the distributional properties of the local linear regression estimator. Applications also include the estimation of econometric (probability-weighted moment, quantile) and biometric (mean remaining lifetime) functions.