A linearization procedure and a VDM/ECM algorithm for penalized and constrained nonparametric maximum likelihood estimation for mixture models
Suppose independent observations Xi X i , i=1,,n i = 1 ,  , n are observed from a mixture model f ( x ; Q ) ? ? f ( x ; ? ) d Q ( ? ) , where ? ? is a scalar and Q(?) Q ( ? ) is a nondegenerate distribution with an unspecified form. We consider to estimate Q(?) Q ( ? ) 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)=g0 g ( Q ) = g 0 . 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 ) - g 0 | 2 under a large penalty factor ?>0 ? > 0 using this algorithm. The algorithm is illustrated with two real data sets.
vdm/ecm nonparametric maximum likelihood mixture models constrained npmle computing algorithm penalized npmle