Smoothing with the Best Rectangle Window is Optimal for All Tapered Rectangle Windows

TL;DR

This paper proves that the best rectangle window is optimal for tapered rectangle window weights in weighted least squares problems, extending to absolute and other loss functions.

Contribution

It demonstrates the optimality of the best rectangle window among tapered rectangle windows for weighted least squares, including extensions to absolute and general loss functions.

Findings

Best rectangle window is optimal for tapered rectangle weights

Optimality extends to least absolute deviations

Results apply to general loss functions

Abstract

We investigate the optimal selection of weight windows for the problem of weighted least squares. We show that weight windows should be symmetric around its center, which is also its peak. We consider the class of tapered rectangle window weights, which are nonincreasing away from the center. We show that the best rectangle window is optimal for such window definitions. We also extend our results to the least absolutes and more general case of arbitrary loss functions to find similar results.