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 tests based on combining p-values. Simulation and empirical studies are conducted for a few settings of the null hypotheses, and they show that methods based on e-processes are efficient.

翻译：本文针对非平稳数据检验条件均值与条件方差的问题展开研究。我们构建了四类非参数复合假设的e值与p值，这些假设在指定均值和方差的同时，还包含数据生成分布形状的其他约束条件。形状约束条件包括对称性、单峰性及其组合形式。利用所获得的e值与p值，我们通过e过程（亦称对赌检验法）构建检验方法，同时采用基于p值合并的检验策略。针对原假设的若干设定场景进行了模拟与实证研究，结果表明基于e过程的方法具有高效性。