Improving estimates of monotone functions by rearrangement

Suppose that a target function is monotonic, namely, weakly increasing, and an original estimate of the target function is available, which is not weakly increasing. Many common estimation methods used in statistics produce such estimates. 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, and the resulting estimate is closer to the true curve in common metrics than the original estimate. We illustrate the results with a computational example and an empirical example dealing with age-height growth charts. Keywords: Monotone function, improved approximation, multivariate rearrangement, univariate rearrangement, growth chart, quantile regression, mean regression, series, locally linear, kernel methods. JEL Classifications: Primary 62G08; Secondary 46F10, 62F35, 62P10.