Improving Point and Interval Estimates of Monotone Functions by Rearrangement

Suppose that a target function f0 : R d → R is monotonic, namely, weakly increasing, and an original estimate f̂ of this target function is available, which is not weakly increasing. Many common estimation methods used in statistics produce such estimates f̂ . 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 f̂, and the resulting estimate is closer to the true curve f0 in common metrics than the original estimate f̂ . 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 f0 with probability 1 − α, 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 f0 with probability greater or equal to 1− α. We illustrate the results with a computational example and an empirical example dealing with age-height growth charts.