We develop and compare e-variables for testing whether $k$ samples of data are drawn from the same distribution, the alternative being that they come from different elements of an exponential family. We consider the GRO (growth-rate optimal) e-variables for (1) a 'small' null inside the same exponential family, and (2) a 'large' nonparametric null, as well as (3) an e-variable arrived at by conditioning on the sum of the sufficient statistics. (2) and (3) are efficiently computable, and extend ideas from Turner et al. [2021] and Wald [1947] respectively from Bernoulli to general exponential families. We provide theoretical and simulation-based comparisons of these e-variables in terms of their logarithmic growth rate, and find that for small effects all four e-variables behave surprisingly similarly; for the Gaussian location and Poisson families, e-variables (1) and (3) coincide; for Bernoulli, (1) and (2) coincide; but in general, whether (2) or (3) grows faster against the small null is family-dependent. We furthermore discuss algorithms for numerically approximating (1).

翻译：本文发展并比较了用于检验$k$个数据样本是否来自同一分布的e变量，其备择假设为样本来自指数族的不同元素。我们考虑以下三种情形的GRO（增长率最优）e变量：（1）在同一指数族内的"小"原假设；（2）非参数化的"大"原假设；以及（3）通过以充分统计量之和为条件构造的e变量。其中（2）和（3）可高效计算，并分别将Turner等[2021]和Wald[1947]的思想从伯努利分布推广至一般指数族。我们从对数增长率角度提供了这些e变量的理论与模拟比较，发现对于小效应量，所有四种e变量的表现惊人地相似；对于高斯位置族和泊松族，e变量（1）与（3）重合；对于伯努利分布，（1）与（2）重合；但一般而言，（2）与（3）中哪个在"小"原假设下增长更快取决于具体族。此外，我们还讨论了数值近似计算（1）的算法。