Hypothesis test for normal mixture models: The EM approach
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正态混合模型的假设检验：EM 方法
- 研究问题
- 面对正态混合模型的似然无界、弱可识别性和混合比例信息量异常，如何进行同质性检验并获得可用的渐近校准？
- 方法与思路
- 作者使用 EM 检验处理有限正态混合的同质性问题，分别研究未知等方差与未知不等方差情形，并开展模拟和遗传学例证。
- 材料陈述的贡献
- 摘要称，EM 检验克服多项模型困难，给出两类方差设定下的极限分布，并称模拟中的渐近近似表现令人满意。
- 阅读关注点
- 应核对 EM 检验的完整构造、参数约束与迭代设置，以及两类方差设定对应的极限分布；具体卡方符号需查原 PDF。
- 阅读边界
- 仅据 OpenAlex 索引摘要，稿件版本未注明。卡方表达式疑有上下标丢失，不能据此重构公式；需查原 PDF，且摘要未提供模拟数值。
原文摘要
Normal mixture distributions are arguably the most important mixture models, and also the most technically challenging. The likelihood function of the normal mixture model is unbounded based on a set of random samples, unless an artificial bound is placed on its component variance parameter. Moreover, the model is not strongly identifiable so it is hard to differentiate between over dispersion caused by the presence of a mixture and that caused by a large variance, and it has infinite Fisher information with respect to mixing proportions. There has been extensive research on finite normal mixture models, but much of it addresses merely consistency of the point estimation or useful practical procedures, and many results require undesirable restrictions on the parameter space. We show that an EM-test for homogeneity is effective at overcoming many challenges in the context of finite normal mixtures. We find that the limiting distribution of the EM-test is a simple function of the 0.5χ02+0.5χ12 and χ12 distributions when the mixing variances are equal but unknown and the χ22 when variances are unequal and unknown. Simulations show that the limiting distributions approximate the finite sample distribution satisfactorily. Two genetic examples are used to illustrate the application of the EM-test.