Several kernel based testing procedures are proposed to solve the problem of model selection in the presence of parameter estimation in a family of candidate models. Extending the two sample test of Gretton et al. (2006), we first provide a way of testing whether some data is drawn from a given parametric model (model specification). Second, we provide a test statistic to decide whether two parametric models are equally valid to describe some data (model comparison), in the spirit of Vuong (1989). All our tests are asymptotically standard normal under the null, even when the true underlying distribution belongs to the competing parametric families.Some simulations illustrate the performance of our tests in terms of power and level.

翻译：针对候选模型族中存在参数估计的模型选择问题，本文提出了若干基于核函数的检验方法。首先，通过扩展Gretton等人（2006）的双样本检验，我们提供了一种检验数据是否来自给定参数模型（模型设定检验）的方法。其次，借鉴Vuong（1989）的思路，我们构建了一个检验统计量，用于判定两个参数模型在描述同一组数据时是否具有同等有效性（模型比较）。所有检验在原假设下均渐近服从标准正态分布，即使真实数据分布属于竞争性参数族也是如此。数值模拟结果展示了所提检验在检验势和显著性水平方面的表现。