cemmap working paper CWP09/07 IMPROVING ESTIMATES OF MONOTONE FUNCTIONS BY REARRANGEMENT

Abstract. 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 f̂∗, 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. Key words. Monotone function, improved approximation, multivariate rearrange-ment, univariate rearrangement, growth chart, quantile regression, mean regression, series, locally linear, kernel method