Minimum profile Hellinger distance estimation for a semiparametric mixture model

Abstract

In this paper, we propose a new effective estimator for a class of semiparametric mixture models where one component has known distribution with possibly unknown parameters while the other component density and the mixing proportion are unknown. Such semiparametric mixture models have been often used in multiple hypothesis testing and the sequential clustering algorithm. The proposed estimator is based on the minimum profile Hellinger distance (MPHD), and its theoretical properties are investigated. In addition, we use simulation studies to illustrate the finite sample performance of the MPHD estimator and compare it with some other existing approaches. The empirical studies demonstrate that the new method outperforms existing estimators when data are generated under contamination and works comparably to existing estimators when data are not contaminated. Applications to two real data sets are also provided to illustrate the effectiveness of the new methodology. The Canadian Journal of Statistics 42: 246–267; 2014 © 2014 Statistical Society of Canada

An Overview of Semiparametric Extensions of Finite Mixture Models

“…This generalization is general enough to include most of the parametric distribution functions, but non-identifiability was the price to pay for such generality. In addition, a two-component mixture of locations model with a known component has been extensively studied during the past decade, by Bordes et al (2006a), Bordes and Vandekerkhove (2010), Patra and Sen (2016), Hohmann and Holzmann (2013), Xiang et al (2014), Ma and Yao (2015), Huang et al (2018), and so on. The model is well motivated and various kinds of estimation methods have been studied.…”

Semiparametric Two-Component Mixture Models When One Component Is Defined Through Linear Constraints

“…The authors suppose that θ is known, f 0 is symmetric around an unknown µ and that r = 1. In [2], the authors studied a more general setup by considering θ unknown and applied model (1) on the Iris data by considering only the first principle component for each observed…”

“…Besides, their asymptotic properties are generally very difficult to establish. Finally, a Hellinger-based two-step directional optimization procedure was proposed in [2]; a first step minimizes the divergence over f 0 and the second step minimizes over the parameters (λ, θ). Their method seems to give good results, but the algorithm is very complicated and no explanation on how to do the calculation is given.…”

“…A comparison between the three methods proposed in [8] and [5] is illustrated in [2] and are comparted to the Hellinger-based iterative method of the same authors on a two-component Gaussian data provided that the parametric component is fully known. As we add θ to the set of unknown parameters, i.e.…”

Maximum likelihood estimation of the mixture of log-concave densities

“…Hunter et al (2007), Bordes et al (2006a), Butucea & Vandekerkhove (2014), and Chee & Wang (2013) considered the extension of (1.1) by assuming all component densities are symmetric but unknown. Bordes et al (2006b), Bordes & Vandekerkhove (2010), Hohmann & Holzmann (2013), Xiang et al (2014), and Ma & Yao (2015) considered the extension of (1.1) when K = 2 and one of the component densities is symmetric but unknown. Mixtures of log-concave densities have been studied by Chang & Walther (2007), Cule et al (2010) and Balabdaoui & Doss (2014).…”