Improving point and interval 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 this 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 by 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. The improvement property of the rearrangement also extends to the construction of confidence bands for monotone functions. Let l and u be the lower and upper endpoint functions of a simultaneous confidence interval [l,u] that covers the true function f0 with probability (1-a), then the rearranged confidence interval [l*,u*], defined by the rearranged lower and upper end-point functions l* and u*, is shorter in length in common norms than the original interval and covers the true function f0 with probability greater or equal to (1-a). We illustrate the results with a computational example and an empirical example dealing with age-height growth charts.