The Asymptotic Equivalence of Level-Based and Share-Based Loss Functions

TL;DR

This paper proves that level-based and share-based loss functions are asymptotically equivalent under certain conditions, implying negligible differences in ratios and ranks when averaged over many units.

Contribution

It establishes the asymptotic equivalence of a broad class of loss functions, specifically weighted exponentiated functions, under decomposable weights.

Findings

Loss functions converge almost surely to a constant ratio.

Differences in ratios and ranks become negligible with large sample sizes.

Numerical and distributive accuracy converge asymptotically.

Abstract

Level-based and share-based loss functions are asymptotically equivalent if, in the limit, their averages converge almost surely to a constant ratio. These loss functions take a target value and its realization as arguments and are often used to measure accuracy. The equivalence is proved for a large class of loss functions, the weighted exponentiated functions, when the weights are decomposable as a particular product form. An upshot is that when losses are averaged for a large number of units, differences in ratios and, hence, ranks, are negligible, when the average (or summed) difference between the target values and their realizations is around zero. This implies the almost sure asymptotic convergence of numerical and distributive accuracy when using these loss functions.