JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. Biometrika Trust is collaborating with JSTOR to digitize, preserve and extend access to Biometrika. SUMMARY Estimation of the optimal linear discriminant function is considered on the basis of a sample of observations known only to belong to a mixture of two univariate normal populations with a common variance. The asymptotic efficiency of the procedure so obtained is evaluated relative to that of Anderson's classification statistic.Some key word8: Asymptotic efficiency; Linear discriminant function; Maximum likelihood estimate; Mixture of normal populations.
Izenman and Sommer (1988) used a non-parametric kernel density estimation technique to fit a seven-component model to the paper thickness of the 1872 Hidalgo stamp issue of Mexico. They observed an apparent conflict when fitting a normal mixture model with three components with unequal variances. This conflict is examined further by investigating the most appropriate number of components when fitting a normal mixture of components with equal variances.
In testing for the number of components in a mixture, it is well known that regularity conditions do not hold for minus twice the log likelihood ratio to have its usual asymptotic null distribution of chi-squared. In an attempt to overcome some of the problems, it has been proposed in the literature that a prior distribution be placed on the mixing proportions. We show here that the usual regularity conditions still do not hold.
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