Improving point and interval estimates of monotone functions by rearrangement

Suppose that a target function ... is monotonic, namely, weakly increasing, and an original estimate of this 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. The improvement property of the rearrangement also extends to the construction of confidence bands for monotone functions. Suppose we have the lower and upper endpoint functions of a simultaneous confidence interval that covers the target function with a pre-specified probability level, then the rearranged confidence interval, defined by the rearranged lower and upper end-point functions, is shorter in length in common norms than the original interval and covers the target function with probability greater or equal to the pre-specified level. We illustrate the results with a computational example and an empirical example dealing with age-height growth charts. Keywords: Monotone function, improved estimation, improved inference, multivariate rearrangement, univariate rearrangement, Lorentz inequalities, growth chart, quantile regression, mean regression, series, locally linear, kernel methods. JEL Classifications: 62G08, 46F10, 62F35, 62P10