There exist some testing procedures based on the maximum mean discrepancy (MMD) to address the challenge of model specification. However, they ignore the presence of estimated parameters in the case of composite null hypotheses. In this paper, we first illustrate the effect of parameter estimation in model specification tests based on the MMD. Second, we propose simple model specification and model selection tests in the case of models with estimated parameters. All our tests are asymptotically standard normal under the null, even when the true underlying distribution belongs to the competing parametric families. A simulation study and a real data analysis illustrate the performance of our tests in terms of power and level.

翻译：现有基于最大均值差异（MMD）的检验方法能够应对模型设定检验的挑战，但这些方法忽略了复合原假设情形下参数估计的存在。本文首先阐明了参数估计对基于MMD的模型设定检验的影响，随后针对含估计参数的模型提出了简洁的模型设定检验与模型选择检验方法。我们提出的所有检验在原假设下均具有渐近标准正态性，即使真实分布属于竞争参数族时亦然。通过模拟研究与实际数据分析，验证了所提检验在功效与水平控制方面的表现。