A constrained maximum likelihood estimation for skew normal mixtures

For a finite mixture of skew normal distributions, the maximum likelihood estimator is not well-defined because of the unboundedness of the likelihood function when scale parameters go to zero and the divergency of the skewness parameter estimates. To overcome these two problems simultaneously, we propose constrained maximum likelihood estimators under constraints on both the scale parameters and the skewness parameters. The proposed estimators are consistent and asymptotically efficient under relaxed constraints on the scale and skewness parameters. Numerical simulations show that in finite sample cases the proposed estimators outperform the ordinary maximum likelihood estimators. Two real datasets are used to illustrate the success of the proposed approach.