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

Suppose independent observations XiXi, i=1,…,ni=1,…,n are observed from a mixture model f(x;Q)≡∫f(x;λ)dQ(λ), where λλ is a scalar and Q(λ)Q(λ) is a nondegenerate distribution with an unspecified form. We consider to estimate Q(λ)Q(λ) by nonparametric maximum likelihood (NPML) method under two scenarios: (1) the likelihood is penalized by a functional g(Q)g(Q); and (2) Q   is under a constraint g(Q)=g0g(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|g(Q)-g0|2 under a large penalty factor γ>0γ>0 using this algorithm. The algorithm is illustrated with two real data sets.