Score-based tests have been used to study parameter heterogeneity across many types of statistical models. This chapter describes a new self-normalization approach for score-based tests of mixed models, which addresses situations where there is dependence between scores. This differs from the traditional score-based tests, which require independence of scores. We first review traditional score-based tests and then propose a new, self-normalized statistic that is related to previous work by Shao and Zhang (2010) and Zhang, Shao, Hayhoe, and Wuebbles (2011). We then provide simulation studies that demonstrate how traditional score-based tests can fail when scores are dependent, and that also demonstrate the good performance of the self-normalized tests. Next, we illustrate how the statistics can be used with real data. Finally, we discuss the potential broad application of self-normalized, score-based tests in mixed models and other models with dependent observations.

翻译：得分检验已被广泛应用于多种统计模型中参数异质性的研究。本章提出了一种新的自归一化方法，用于混合模型的得分检验，以应对得分之间存在相关性的情形。这与传统的得分检验不同——后者要求得分具有独立性。我们首先回顾传统得分检验，随后提出一种新的自归一化统计量，该统计量与Shao和Zhang（2010）以及Zhang、Shao、Hayhoe和Wuebbles（2011）的前期工作相关。接着，我们通过模拟研究证明：当得分存在相关性时，传统得分检验可能失效，而自归一化检验则表现良好。然后，我们展示如何将这些统计量应用于实际数据。最后，我们讨论自归一化得分检验在混合模型及其他具有相关观测的模型中潜在的广泛应用前景。