Abstract
Location mixture models, resulting in shifting a common distribution with some probability, have been widely used to account for existence of clusters in the data. Assuming only symmetry of this common distribution allows for great flexibility, especially when the traditional normality assumption is violated. This semi-parametric model has been studied in several papers, where the mixture parameters are first estimated before constructing an estimator for the non-parametric component. The plug-in method suggested by Hunter et al. (2007) has the merit to be easily implementable and fast to compute. However, no result is available on the limit distribution of the obtained estimator, hindering for instance construction of asymptotic confidence intervals. In this paper, we give sufficient conditions on the symmetric distribution for asymptotic normality to hold. In case the symmetric distribution admits a log-concave density, our assumptions are automatically satisfied. The obtained result has to be used with caution in case the mixture location are too close or the mixing probability is close to $0$ or $1$. Three examples are considered where we show that the estimator is not to be advocated when the mixture components are not well separated.
Highlights
When testing for presence of mixing, the numerical results obtained in Balabdaoui and Doss (2016) for the asymptotic power show the higher performance of the symmetric log-concave maximum likelihood estimator (MLE) when compared to the Gaussian one
Hunter et al √(2007) give in their Theorem 4 conditions under which θn converges at the n-rate to a 3-dimensional centered Gaussian distribution with a dispersion matrix given by J−1ΣJ−1
We show that θn is asymptotically normal under some sufficient conditions
Summary
This model is semi-parametric since the unknown parameters are the 3-dimensional vector (π, μ1, μ2) and the symmetric distribution G It has been considered by several authors, e.g. Bordes et al (2006), Hunter et al (2007), Chee and Wang (2013), Butucea and Vandekerkhove (2014), and more recently Balabdaoui and Doss (2016). The authors showed that their estimator is consistent and established that it is asymptotically normal, under the assumption that the mixture model is identifiable They used the identifiability conditions found by Hunter et al (2007), which we discuss. The only distributions which are not 2-identifiable are those which are themselves symmetric; see their Theorem 2 This result is proved by first showing that a necessary and sufficient condition for the vector parameter to yield a 2-identifiable mixed distribution given by (1.1) is that. Note that the latter is nothing but saying that the L2 distance between F 0 Δ(π0, −μ01, −μ02) and (1 − F 0(−·)) Δ(π0, μ01, μ02) is equal to 0
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