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

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.