Limiting processes for divergence measures beyond I-divergence

Determine which large-sample quasi-independent sampling processes, if any, lead to asymptotic conditional sampling probabilities governed by divergence measures other than the I-divergence (Kullback–Leibler divergence) within the Total Empiricism (Tot) framework, thereby justifying minimization of such alternative divergences for statistical hypothesis testing.

Background

The paper introduces the Total Empiricism (Tot) statistical framework and the Information-test (I-test), which relies on minimization of the I-divergence (Kullback–Leibler divergence) and its large-sample universality for quasi-independent sampling processes. In this framework, the conditional sampling probability concentrates around the I-projection, leading to a universal chi-squared distribution for the pivotal statistic.

While the authors justify the I-divergence through multiple sampling processes whose large-N limits are governed by it, they highlight a gap regarding other divergence measures: it remains unknown what limiting sampling processes would correspond to these alternative divergences. Resolving this would broaden Tot by establishing when and why other divergences are appropriate for hypothesis testing.