We address the problem of testing conditional mean and conditional variance for non-stationary data. We build e-values and p-values for four types of non-parametric composite hypotheses with specified mean and variance as well as other conditions on the shape of the data-generating distribution. These shape conditions include symmetry, unimodality, and their combination. Using the obtained e-values and p-values, we construct tests via e-processes, also known as testing by betting, as well as some tests based on combining p-values for comparison. Although we mainly focus on one-sided tests, the two-sided test for the mean is also studied. Simulation and empirical studies are conducted under a few settings, and they illustrate features of the methods based on e-processes.

翻译：我们研究了非平稳数据条件下条件均值与条件方差的检验问题。针对四类具有指定均值、方差及数据生成分布形态约束的非参数复合假设，我们构建了e值与p值。这些形态约束包括对称性、单峰性及其组合。利用所得到的e值与p值，我们通过e过程（亦称"赌注检验"）构建检验方法，并对比了基于p值组合的若干检验方案。尽管主要关注单侧检验，本文亦研究了均值的双侧检验方法。通过多组模拟与实证研究，我们展示了基于e过程的检验方法特性。