Fitting a Mixture Distribution to a Variable Subject to Heteroscedastie Measurement Errors
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In a structural errors-in-variables model the true regressors are treated as stochastic variables that can only be measured with an additional error. Therefore the distribution of the latent predictor variables and the distribution of the measurement errors play an important role in the analysis of such models. In this article the conventional assumptions of normality for these distributions are extended in two directions. The distribution of the true regressor variable is assumed to be a mixture of normal distributions and the measurement errors are again taken to be normally distributed but the error variances are allowed to be heteroscedastie. It is shown how an EM algorithm solely based on the error-prone observations of the latent variable can be used to find approximate ML estimates of the distribution parameters of the mixture. The procedure is illustrated by a Swiss data set that consists of regional radon measurements. The mean concentrations of the regions serve as proxies for the true regional averages of radon. The different variability of the measurements within the regions motivated this approach.
KeywordsHeteroscedastie measurement errors Finite mixture distribution EM algorithm
This research was partly supported by the Deutsche Forschungsgemeinschaft (German Research Council). I would like to thank Ch. E. Minder for discussion and introducing the problem. I would also like to thank an anonymous referee for directing my attention to some very general problems in the estimation of mixture models. Helpful discussions with H. Schneeweiss and R. Wolf are gratefully acknowledged.
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