Improving Point and Interval Estimates of Monotone Functions by Rearrangement

TL;DR

This paper introduces rearrangement techniques to improve monotone function estimates and their confidence intervals, ensuring they are closer to the true function and shorter in length while maintaining coverage probability.

Contribution

It develops a method to transform non-monotonic estimates and confidence intervals into monotonic ones that are more accurate and efficient.

Findings

Rearranged estimates are closer to the true monotone function.

Rearranged confidence intervals are shorter and maintain coverage.

Method demonstrated with an age-height growth chart example.

Abstract

Suppose that a target function is monotonic, namely, weakly increasing, and an available original estimate of this target function is not weakly increasing. Rearrangements, univariate and multivariate, transform the original estimate to a monotonic estimate that always lies closer in common metrics to the target function. Furthermore, suppose an original simultaneous confidence interval, which covers the target function with probability at least $1-\alpha$, is defined by an upper and lower end-point functions that are not weakly increasing. Then the rearranged confidence interval, defined by the rearranged upper and lower end-point functions, is shorter in length in common norms than the original interval and also covers the target function with probability at least $1-\alpha$. We demonstrate the utility of the improved point and interval estimates with an age-height growth chart example.