The Asymptotic Equivalence of Level-Based and Share-Based Loss Functions
Charles D. Coleman
TL;DR
The paper proves that for a broad class of weighted exponentiated loss functions, the mean loss computed with level arguments and that computed with share arguments converge almost surely to a fixed ratio, making numeric and distributive accuracy asymptotically equivalent. Using a Cesàro-mean framework and exchangeability (plus sparse deviation conditions), it derives a key level–share identity and establishes convergence of the ratio with constants $K=\mu_x^p$ or $\mu_y^p$ depending on the weighting. It provides explicit constants for standard losses, derives a convergence-rate for the ratio, and demonstrates that asymptotic equivalence does not guarantee small-sample equivalence, supported by an empirical census example. The results imply that, in large datasets, the choice between level or share arguments is asymptotically moot for these loss measures, though dissimilarity indices may lose interpretability in finite samples.
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.