Optimization by Vector Space Methods
Optimization by Vector Space Methods
Asymptotic theory for maximum likelihood in nonparametric mixture models
Computational Statistics & Data Analysis
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Suppose independent observations X"i, i=1,...,n are observed from a mixture model f(x;Q)=@!f(x;@l)dQ(@l), where @l is a scalar and Q(@l) is a nondegenerate distribution with an unspecified form. We consider to estimate Q(@l) 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 @c0 using this algorithm. The algorithm is illustrated with two real data sets.