We propose novel kernel-based tests for assessing the equivalence between distributions. Traditional goodness-of-fit testing is inappropriate for concluding the absence of distributional differences, because failure to reject the null hypothesis may simply be a result of lack of test power, also known as the Type-II error. This motivates \emph{equivalence testing}, which aims to assess the \emph{absence} of a statistically meaningful effect under controlled error rates. However, existing equivalence tests are either limited to parametric distributions or focus only on specific moments rather than the full distribution. We address these limitations using two kernel-based statistical discrepancies: the \emph{kernel Stein discrepancy} and the \emph{Maximum Mean Discrepancy}. The null hypothesis of our proposed tests assumes the candidate distribution differs from the nominal distribution by at least a pre-defined margin, which is measured by these discrepancies. We propose two approaches for computing the critical values of the tests, one using an asymptotic normality approximation, and another based on bootstrapping. Numerical experiments are conducted to assess the performance of these tests.

翻译：我们提出了一种基于核函数的新型检验方法，用于评估分布间的等价性。传统的拟合优度检验不适用于推断分布差异不存在的情况，因为未能拒绝原假设可能仅仅是检验功效不足（即第二类错误）的结果。这促使了**等价性检验**的发展，其目标是在控制错误率的前提下评估统计意义上显著效应的**不存在性**。然而，现有的等价性检验要么局限于参数分布，要么仅关注特定矩而非完整分布。我们利用两种基于核函数的统计差异度量——**核斯坦因差异**和**最大均值差异**——来解决这些局限性。我们所提出检验的原假设假定候选分布与名义分布之间的差异至少超过一个预定义的边界值，该边界值由上述差异度量进行量化。我们提出了两种计算检验临界值的方法：一种采用渐近正态性近似，另一种基于自助法。数值实验被用于评估这些检验的性能。