Information projections have found important applications in probability theory, statistics, and related areas. In the field of hypothesis testing in particular, the reverse information projection (RIPr) has recently been shown to lead to growth-rate optimal (GRO) e-statistics for testing simple alternatives against composite null hypotheses. However, the RIPr as well as the GRO criterion are undefined whenever the infimum information divergence between the null and alternative is infinite. We show that in such scenarios, under some assumptions, there still exists a measure in the null that is closest to the alternative in a specific sense. Whenever the information divergence is finite, this measure coincides with the usual RIPr. It therefore gives a natural extension of the RIPr to certain cases where the latter was previously not defined. This extended notion of the RIPr is shown to lead to optimal e-statistics in a sense that is a novel, but natural, extension of the GRO criterion. We also give conditions under which the (extension of the) RIPr is a strict sub-probability measure, as well as conditions under which an approximation of the RIPr leads to approximate e-statistics. For this case we provide tight relations between the corresponding approximation rates.

翻译：信息投影在概率论、统计学及相关领域已展现出重要应用价值。尤其在假设检验领域，反向信息投影（RIPr）最近被证明可为简单备择假设对复合零假设的检验问题导出增长率最优（GRO）的e统计量。然而，当零假设与备择假设之间的信息散度下确界为无穷大时，RIPr及GRO准则均无定义。本文证明在此类情形下，在某些假设条件下，仍存在零假设中的某个测度在特定意义上最接近备择假设。当信息散度为有限值时，该测度与经典RIPr一致。因此，这为RIPr提供了一种自然的扩展，使其适用于原先未定义的特定场景。研究进一步表明，这种扩展的RIPr概念能导出某种意义下的最优e统计量，该意义是GRO准则的一种新颖而自然的扩展。我们还给出了（扩展的）RIPr成为严格次概率测度的条件，以及RIPr近似导致近似e统计量的条件。针对近似情形，我们建立了相应近似率之间的紧致关系。