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A likelihood ratio test of a homoscedastic normal mixture against a heteroscedastic normal mixture

机译：均方差正常混合物与异方差正常混合物的似然比检验

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摘要

It is generally assumed that the likelihood ratio statistic for testing the null hypothesis that data arise from a homoscedastic normal mixture distribution versus the alternative hypothesis that data arise from a heteroscedastic normal mixture distribution has an asymptotic χ~2 reference distribution with degrees of freedom equal to the difference in the number of parameters being estimated under the alternative and null models under some regularity conditions. Simulations show that the χ~2 reference distribution will give a reasonable approximation for the likelihood ratio test only when the sample size is 2000 or more and the mixture components are well separated when the restrictions suggested by Hathaway (Ann. Stat. 13:795-800, 1985) are imposed on the component variances to ensure that the likelihood is bounded under the alternative distribution. For small and medium sample sizes, parametric bootstrap tests appear to work well for determining whether data arise from a normal mixture with equal variances or a normal mixture with unequal variances.
机译：通常假设用于检验零假设的数据的似然比统计量来自同分正态正态混合分布，而替代假设该数据来自异方差正态混合分布具有自由度等于的渐近的χ〜2参考分布。在某些规则性条件下，在替代模型和空模型下估计的参数数量差异。模拟显示，只有当样本量为2000或更大，并且当Hathaway建议的限制条件下，混合物成分充分分离时，χ〜2参考分布才能为似然比测试提供合理的近似值（Ann。Stat。13：795- 800，1985）应用于成分方差，以确保可能性在替代分布的范围内。对于中小型样本，参数自举测试似乎可以很好地确定数据是来自方差相等的正态混合物还是方差不相等的正态混合物。

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