Bias from misspecification of the component variances in a normal mixture

Bias in parameter estimates can be substantial when heteroscedastic normal mixtures are misspecified as homoscedastic normal mixtures, and vice versa. We show through simulations that the maximum likelihood estimators under the false assumption of equal variances are inconsistent and bias in parameter estimates is appreciable and even substantial when the mixture components are not well-separated. Finite sample bias in parameter estimates is close to the asymptotic bias even for a sample size of 200 or less. When homoscedastic normal mixtures are misspecified as heteroscedastic normal mixtures, the maximum likelihood estimators are consistent. However, the maximum likelihood estimators under a correctly specified homoscedastic mixture model converge to the true parameter values faster than those under a misspecified heteroscedastic mixture model. The bias of the maximum likelihood estimators is less dependent on the lower bound imposed on the component variances to ensure that the likelihood is bounded under the false assumption of unequal variances when the sample size is 500 or more and the component distributions are well-separated. An example is given to demonstrate the effects of a misspecification of the component variances on estimates of the prevalence of hypertension using normal mixtures.