By G. J. McLachlan, K. E. Basford, Geoffrey J. McLachlan

ISBN-10: 0824776917

ISBN-13: 9780824776916

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**Extra info for Mixture Models**

**Example text**

Kiefer (1978) verified for a mixture of univariate normal distributions in the more general case of the switching regression model that there is a sequence of roots of the likelihood equation which is consistent and asymp totically efficient and normally distributed. With probability tending to one, these roots correspond to local maxima in the interior of the param eter space; see also Peters and Walker (1978). This consistent sequence of roots is essentially unique. The reader is referred to Huzurbazar (1948) and Perlman (1983) for a precise statement of the uniqueness of a consistent sequence of roots of the likelihood equation.

If the mixing proportions 7t¿ are taken to be equal, or equivalently a separate sampling scheme is adopted for the data, then the |W | criterion is obtained, as originally suggested by Friedman and Rubin (1967). 2) appears to go some way to overcoming this. If we further assume that the common within population covariance matrix is spherical (S = cr^I, I the identity matrix), then the classification likelihood approach yields the trace W criterion, as proposed by Edwards and Cavalli-Sforza (1965) for partitioning the data in a hierarchical manner.

8 , the nonexistence of the maximum likelihood estimate does not place a caveat on proceedings, as the essential aim of likelihood estimation*, is to find a sequence of roots of the likelihood equation which is consistent, and hence efficient if the usual regularity conditions hold. However, with mixture models the likelihood equation will generally have multiple roots. Thus even if it is known that there exists a sequence of roots of the likeli hood equation with the desired asymptotic properties, there is the problem of identifying this sequence.

### Mixture Models by G. J. McLachlan, K. E. Basford, Geoffrey J. McLachlan

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