|
Abstract:
|
Due to the unboundedness of likelihood function, the statistical inference based on data from the finite mixture of normal distributions in both mean and variance is particularly difficult. Through the use of some suitable penalized likelihood functions, we develop a new method applicable to both univariate and multivariate normal mixture models. We show that the penalized maximum likelihood estimators (PMLE) are strongly consistent, asymptotically normal, and invariant under linear transformations. We also show that the constrained maximum likelihood estimator, as proposed in Hathaway (1985), is still consistent when the lower bound $c$ placed on the ratio of any two component variances satisfies that $c \geq \exp \{-k (\log^2n)\}, ~ \forall ~ k>0$. Furthermore, this approach can be readily extended to solve similar problems in other mixture models.
|
