## Abstract

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.

Original language | English (US) |
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Pages (from-to) | 2946-2957 |

Number of pages | 12 |

Journal | Computational Statistics and Data Analysis |

Volume | 51 |

Issue number | 6 |

DOIs | |

State | Published - Mar 1 2007 |

## Keywords

- Computing algorithm
- Constrained NPMLE
- Mixture models
- Nonparametric maximum likelihood
- Penalized NPMLE
- VDM/ECM

## ASJC Scopus subject areas

- Statistics and Probability
- Computational Mathematics
- Computational Theory and Mathematics
- Applied Mathematics