A linearization procedure and a VDM/ECM algorithm for penalized and constrained nonparametric maximum likelihood estimation for mixture models - Wang, Ji-Ping (Computational statistics & data analysis, 2007)
Suppose independent observations X i , i = 1 , … , n are observed from a mixture model f ( x ; Q ) ≡ ∫ f ( x ; λ ) d Q ( λ ) , 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 ) = 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 ) - g 0 | 2 under a large penalty factor γ > 0 using this algorithm. The algorithm is illustrated with two real data sets.