Computing the Standards Errors of Mixture Model Parameters with EM when Classes are Well Separated
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It is shown that for finite mixtures the missing information tends to zero as the number of observations on each subject increases. Then, the classes become perfectly separated (i.e. the posterior membership probabilities are close to 0 or 1), the observed information tends to the complete information and the class-specific parameters in the mixture model become information orthogonal across classes. Then the asymptotic standard errors of parameter estimates can be obtained directly from the EM algorithm. The degree of class-separation is derived for which the amount of missing observation is approximately negligible and the asymptotic standard errors based on the complete information matrix are sufficiently accurate. Empirical illustrations are provided. A Monte Carlo study is performed to examine the extent to which the approximation is adequate. A comparison is made with other methods to approximate the observed information matrix. It is concluded that if the entropy of the posterior probabilities is larger than 0.95 the proposed approximation is reasonably accurate.
Key WordsMixture Models EM algorithm Observed Information Complete Information Missing Information Information Orthogonality Entropy
The author is greatly indebted to Paul Bekker, Frenkel ter Hofstede, Ton Steerneman and two anonymous reviewers for their very useful comments and suggestions.
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