A linearization procedure and a VDM/ECM algorithm for penalized and constrained nonparametric maximum likelihood estimation for mixture models

Suppose independent observations X_i, i=1,…,n are observed from a mixture model , where λ is a scalar and Q(λ) is a nondegenerate distribution with an unspecified form. We consider to estimate Q(λ) by nonparametric maximum likelihood (NPML) method under two scenarios: (1) the likelihood is penalized by a functional g(Q); and (2) Q is under a constraint g(Q)=g₀. We propose a simple and reliable algorithm termed VDM/ECM for Q-estimation when the likelihood is penalized by a linear functional. We show this algorithm can be applied to a more general situation where the penalty is not linear, but a function of linear functionals by a linearization procedure. The constrained NPMLE can be found by penalizing the quadratic distance |g(Q)-g₀|² under a large penalty factor γ>0 using this algorithm. The algorithm is illustrated with two real data sets.