Valid sequential inference on probability forecast performance

Abstract

Probability forecasts for binary events play a central role in many applications. Their quality is commonly assessed with proper scoring rules, which assign forecasts numerical scores such that a correct forecast achieves a minimal expected score. In this paper, we construct e-values for testing the statistical significance of score differences of competing forecasts in sequential settings. E-values have been proposed as an alternative to $p$-values for hypothesis testing, and they can easily be transformed into conservative $p$-values by taking the multiplicative inverse. The e-values proposed in this article are valid in finite samples without any assumptions on the data-generating processes. They also allow optional stopping, so a forecast user may decide to interrupt evaluation, taking into account the available data at any time, and still draw statistically valid inference, which is generally not true for classical p-value-based tests. In a case study on post-processing of precipitation forecasts, state-of-the-art forecast dominance tests and e-values lead to the same conclusions.