DAOUIA, A. and B.U. PARK On Projection-Type Estimators of Multivariate Isotonic Functions

Let M be an isotonic real-valued function on a compact subset of Rd and let M̂n be an unconstrained estimator of M. A feasible monotonizing technique is to take the largest (smallest) monotone function that lies below (above) the estimator M̂n or any convex combination of these two envelope estimators. When the process rn(M̂n−M) is asymptotically equicontinuous for some sequence rn> 0, we show that these projected estimators are rn-equivalent in probability to the original unrestricted estimator. Our first motivating application involves a monotone estimator of the conditional distribu-tion function that has the distributional properties of the local linear regression esti-mator. Applications also include the estimation of econometric (probability-weighted moment, quantile-based) and biometric (mean remaining lifetime) functions