Breakdown points for maximum likelihood estimators of location–scale mixtures
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位置—尺度混合模型极大似然估计的崩溃点
- 研究问题
- 聚类分析中，正态混合、t 混合和附加噪声成分的方法面对极端离群点时有多稳健，固定或估计簇数如何改变崩溃行为？
- 方法与思路
- 作者讨论适合聚类的稳健性度量，推导几类混合模型的崩溃点界，并比较固定簇数、估计簇数及隔离极端离群点的作用。
- 材料陈述的贡献
- 摘要称，t 混合和附加噪声成分虽改善中等离群点下的稳定性，崩溃表现并未显著优于正态混合；另提出非正规均匀噪声混合。
- 阅读关注点
- 应核对崩溃点定义、参数发散的判据、固定簇数时的例外情形，以及估计簇数和使用非正规均匀成分的约束。
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- 仅据版本未注明的 OpenAlex 摘要；崩溃点界及拟议方法的条件未完整提供，抗极端污染的结论不能替代一般聚类准确性比较。
原文摘要
ML-estimation based on mixtures of Normal distributions is a widely used tool for cluster analysis. However, a single outlier can make the parameter estimation of at least one of the mixture components break down. Among others, the estimation of mixtures of t-distributions by McLachlan and Peel [Finite Mixture Models (2000) Wiley, New York] and the addition of a further mixture component accounting for “noise” by Fraley and Raftery [The Computer J. 41 (1998) 578–588] were suggested as more robust alternatives. In this paper, the definition of an adequate robustness measure for cluster analysis is discussed and bounds for the breakdown points of the mentioned methods are given. It turns out that the two alternatives, while adding stability in the presence of outliers of moderate size, do not possess a substantially better breakdown behavior than estimation based on Normal mixtures. If the number of clusters s is treated as fixed, r additional points suffice for all three methods to let the parameters of r clusters explode. Only in the case of r=s is this not possible for t-mixtures. The ability to estimate the number of mixture components, for example, by use of the Bayesian information criterion of Schwarz [Ann. Statist. 6 (1978) 461–464], and to isolate gross outliers as clusters of one point, is crucial for an improved breakdown behavior of all three techniques. Furthermore, a mixture of Normals with an improper uniform distribution is proposed to achieve more robustness in the case of a fixed number of components.