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Nonparametric estimation of component distributions in a multivariate mixture. (English) Zbl 1018.62021
Summary: Suppose $$k$$-variate data are drawn from a mixture of two distributions, each having independent components. It is desired to estimate the univariate marginal distributions in each of the products, as well as the mixing proportion. This is the setting of two-class, fully parametrized latent models that has been proposed for estimating the distributions of medical test results when disease status is unavailable. The problem is one of inference in a mixture of distributions without training data, and until now it has been tackled only in a fully parametric setting.
We investigate the possibility of using nonparametric methods. Of course, when $$k=1$$ the problem is not identifiable from a nonparametric viewpoint. We show that the problem is “almost” identifiable when $$k=2$$; there, the set of all possible representations can be expressed, in terms of any one of those representations, as a two-parameter family. Furthermore, it is proved that when $$k\geq 3$$ the problem is nonparametrically identifiable under particularly mild regularity conditions. In this case we introduce root-$$n$$ consistent nonparametric estimators of the $$2k$$ univariate marginal distributions and the mixing proportion. Finite-sample and asymptotic properties of the estimators are described.

##### MSC:
 62G05 Nonparametric estimation 62P10 Applications of statistics to biology and medical sciences; meta analysis 62G07 Density estimation 62H12 Estimation in multivariate analysis
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