We provide a general condition under which e-variables in the form of a simple-vs.-simple likelihood ratio exist when the null hypothesis is a composite, multivariate exponential family. Such `simple' e-variables are easy to compute and expected-log-optimal with respect to any stopping time. Simple e-variables were previously only known to exist in quite specific settings, but we offer a unifying theorem on their existence for testing exponential families. We start with a simple alternative $Q$ and a regular exponential family null. Together these induce a second exponential family ${\cal Q}$ containing $Q$, with the same sufficient statistic as the null. Our theorem shows that simple e-variables exist whenever the covariance matrices of ${\cal Q}$ and the null are in a certain relation. Examples in which this relation holds include some $k$-sample tests, Gaussian location- and scale tests, and tests for more general classes of natural exponential families.

翻译：我们给出了一个通用条件，在该条件下，当原假设为复合多元指数族时，形如简单对简单似然比的E变量存在。这类"简单"E变量易于计算，且在任何停止时间下均为期望对数最优的。此前，简单E变量仅被证实在相当特定的设定中存在，而本文为检验指数族时其存在性提供了一个统一性定理。我们从简单备择假设$Q$和正则指数族原假设出发，二者共同诱导出第二个包含$Q$的指数族${\cal Q}$，该族与原假设具有相同的充分统计量。本定理表明，当${\cal Q}$与原假设的协方差矩阵满足特定关系时，简单E变量总是存在。满足该关系的示例包括某些$k$样本检验、高斯位置检验与尺度检验，以及对更一般类自然指数族的检验。