IMPROVING ESTIMATES OF MONOTONE FUNCTIONS BY REARRANGEMENT

Suppose that a target function f0 : Rd ! R is monotonic, namely, weakly increasing, and an original estimate ^ f of the target function is available, which is not weakly increasing. Many common estimation methods used in statistics produce such estimates ^ f. We show that these estimates can always be improved with no harm using rearrangement techniques: The rearrangement methods, univariate and multivariate, transform the original estimate to a monotonic estimate ^ f¤, and the resulting estimate is closer to the true curve f0 in common metrics than the original estimate ^ f. We illustrate the results with a computational example and an empirical example dealing with age-height growth charts.