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Two-component mixtures with independent coordinates as conditional mixtures: nonparametric identification and estimation. (English) Zbl 1337.62056
Summary: We show how the multivariate two-component mixtures with independent coordinates in each component by P. Hall and X.-H. Zhou [Ann. Stat. 31, No. 1, 201–224 (2003; Zbl 1018.62021)] can be studied within the framework of conditional mixtures as recently introduced by M. Henry et al. [“Identifying finite mixtures in econometric models”, Cowles Foundation Discussion Paper Series 1767 (2010), http://cowles.yale.edu/sites/default/files/files/pub/d17/d1767.pdf]. Here, the conditional distribution of the random variable $$Y$$ given the vector of regressors $$Z$$ can be expressed as a two-component mixture, where only the mixture weights depend on the covariates. Under appropriate tail conditions on the characteristic functions and the distribution functions of the mixture components, which allow for flexible location-scale type mixtures, we show identification and provide asymptotically normal estimators. The main application for our results are bivariate two-component mixtures with independent coordinates, the case not previously covered by Hall and Zhou [loc. cit.]. In a simulation study we investigate the finite-sample performance of the proposed methods. The main new technical ingredient is the estimation of limits of quotients of two characteristic functions in the tails from independent samples, which might be of some independent interest.

##### MSC:
 62G05 Nonparametric estimation 62H12 Estimation in multivariate analysis 62G20 Asymptotic properties of nonparametric inference
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