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：本文针对非平稳数据的条件均值与条件方差检验问题展开研究。针对四类具有指定均值与方差、且对数据生成分布形态存在其他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、单峰性及其组合形式。利用所得e值与p值，我们通过e过程（亦称博弈检验法）构建检验方法，同时基于p值组合构建若干对比检验。尽管主要关注单边检验，本文亦探讨了均值的双边检验问题。通过多种设定下的模拟与实证研究，揭示了基于e过程的检验方法的特性。