Improving Point and Interval Estimates of Monotone Functions by Rearrangement

Suppose that a target function $f_0: \Bbb{R}^d \to \Bbb{R}$ is monotonic, namely, weakly increasing, and an original estimate $\hat f$ of this target function is available, which is not weakly increasing. Many common estimation methods used in statistics produce such estimates $\hat f$. 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 $\hat f^*$, and the resulting estimate is \textit{closer} to the true curve $f_0$ in common metrics than the original estimate $\hat f$. The improvement property of the rearrangement also extends to the construction of confidence bands for monotone functions. Let $\ell$ and $u$ be the lower and upper endpoint functions of a simultaneous confidence interval $[\ell, u]$ that covers $f_0$ with probability $1-\alpha$, then the rearranged confidence interval $[\ell^*, u^*]$, defined by the rearranged lower and upper end-point functions $\ell^*$ and $u^*$, is shorter in length in common norms than the original interval and covers $f_0$ with probability greater or equal to $1-\alpha$. We illustrate the results with a computational example and an empirical example dealing with age-height growth charts.