Consistency of The Constrained Maximum Likelihood Estimator in Finite Normal Mixture Models

Due to non-regularity of the finite mixture of normal dis-tributions in both mean and variance, the usual maxi-mum likelihood estimator is not well defined for this most important class of mixture models. By requiring the ra-tio of the variances of any two component distributions being larger than a constant c not depending on sample size, Hathaway (1985) established the consistency of the constrained MLE of the mixing distribution. This result, however, can be void when the true distribution does not satisfy the constraint. Thus, it is desirable to let the con-stant c decreases as the sample size increases to ∞. Two interesting research problems are to determine whether the consistency remains true when cn decreases to 0 and if so, how fast it is allowed to decrease to 0 as the sample size increases. In this note, we prove the consistency of the new constrained maximum likelihood estimator when cn is a decreasing sequence satisfying log cn> −k(log2 n) for any k> 0