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. A prime example in which this relation holds is testing whether a parameter in a linear regression is 0. Other examples include some $k$-sample tests, Gaussian location- and scale tests, and tests for more general classes of natural exponential families. While in all these examples, the implicit composite alternative is also an exponential family, in general this is not required.

翻译：我们提出了一个一般性条件，当零假设为复合多元指数族时，存在以简单对简单似然比形式表示的e变量。此类"简单"e变量易于计算，且对于任意停止时间均具有期望对数最优性。此前已知简单e变量仅存在于特定场景中，而本文提供了关于指数族检验中其存在性的统一定理。我们从简单备择假设$Q$与正则指数族零假设出发，二者共同导出一个包含$Q$的第二指数族${\cal Q}$，且其充分统计量与零假设相同。我们的定理证明，当${\cal Q}$与零假设的协方差矩阵满足特定关系时，简单e变量必然存在。该关系成立的典型示例包括线性回归中参数是否为零的检验。其他示例涵盖部分$k$样本检验、高斯位置与尺度检验，以及更广义自然指数族类别的检验。尽管这些示例中隐含的复合备择假设同样是指数族，但该条件在一般情况下并非必需。