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Testing the Order of Multivariate Normal Mixture Models

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Document pages: 54 pages

Abstract: Finite mixtures of multivariate normal distributions have been widely used inempirical applications in diverse fields such as statistical genetics andstatistical finance. Testing the number of components in multivariate normalmixture models is a long-standing challenge even in the most important case oftesting homogeneity. This paper develops likelihood-based tests of the nullhypothesis of $M 0$ components against the alternative hypothesis of $M 0 + 1$components for a general $M 0 geq 1$. For heteroscedastic normal mixtures, wepropose an EM test and derive the asymptotic distribution of the EM teststatistic. For homoscedastic normal mixtures, we derive the asymptoticdistribution of the likelihood ratio test statistic. We also derive theasymptotic distribution of the likelihood ratio test statistic and EM teststatistic under local alternatives and show the validity of parametricbootstrap. The simulations show that the proposed test has good finite samplesize and power properties.

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