On the consistency of penalized MLEs for Erlang mixtures

In Yin and Lin (2016), a new penalty, termed as iSCAD penalty, is proposed to obtain the maximum likelihood estimates (MLEs) of the weights and the common scale parameter of an Erlang mixture model. In that paper, it is shown through simulation studies and a real data application that the penalty provides an efficient way to determine the MLEs and the order of the mixture. In this paper, we provide a theoretical justification and show that the penalized maximum likelihood estimators of the weights and the scale parameter as well as the order of mixture are all consistent.