The testing problem for the order of finite mixture models has a long history and remains an active research topic. Since Ghosh & Sen (1985) revealed the hard‐to‐manage asymptotic properties of the likelihood ratio test, many successful alternative approaches have been developed. The most successful attempts include the modified likelihood ratio test and the EM‐test, which lead to neat solutions for finite mixtures of univariate normal distributions, finite mixtures of single‐parameter distributions, and several mixture‐like models. The problem remains challenging, and there is still no generic solution for location‐scale mixtures. In this article, we provide an EM‐test solution for homogeneity for finite mixtures of location‐scale family distributions. This EM‐test has nonstandard limiting distributions, but we are able to find the critical values numerically. We use computer experiments to obtain appropriate values for the tuning parameters. A simulation study shows that the fine‐tuned EM‐test has close to nominal type I errors and very good power properties. Two application examples are included to demonstrate the performance of the EM‐test.

中文翻译：

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有限位置比例混合物下的均质性测试

有限混合模型阶的测试问题历史悠久，仍然是一个活跃的研究主题。自从Ghosh＆Sen（1985）揭示了似然比检验的难以管理的渐近性质以来，已经开发了许多成功的替代方法。最成功的尝试包括修改后的似然比检验和EM-test，它们为单变量正态分布的有限混合，单参数分布的有限混合以及几种类似混合的模型提供了简洁的解决方案。该问题仍然具有挑战性，并且对于位置范围的混合物仍然没有通用的解决方案。在本文中，我们提供了针对位置尺度族分布的有限混合的同质性的EM测试解决方案。该EM测试具有非标准的限制分布，但是我们能够通过数字找到临界值。我们使用计算机实验来获得调整参数的适当值。仿真研究表明，经过微调的EM测试具有接近标称的I型误差和非常好的功率特性。其中包括两个应用示例，以演示EM测试的性能。