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August 2004 Breakdown points for maximum likelihood estimators of location–scale mixtures
Christian Hennig
Ann. Statist. 32(4): 1313-1340 (August 2004). DOI: 10.1214/009053604000000571


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.


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Christian Hennig. "Breakdown points for maximum likelihood estimators of location–scale mixtures." Ann. Statist. 32 (4) 1313 - 1340, August 2004.


Published: August 2004
First available in Project Euclid: 4 August 2004

zbMATH: 1047.62063
MathSciNet: MR2089126
Digital Object Identifier: 10.1214/009053604000000571

Primary: 62F35
Secondary: 62H30

Keywords: classification breakdown point , mixtures of t-distributions , Model-based cluster analysis , noise component , normal mixtures , robust statistics

Rights: Copyright © 2004 Institute of Mathematical Statistics

Vol.32 • No. 4 • August 2004
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