Nonparametric regression estimators for mixtures with varying concentrations

Abstract: Finite mixture models naturally arise in statistical analysis of biological and sociological data. If the sub-population which a subject belongs to is not known exactly, the distribution of its variables is a mixture of the sub-populations’ distributions. In the classical finite mixture models (FMM) the concentrations of the components in the mixture (mixing probabilities) are the same for all observations. In a more flexible mixture with varying concentrations model (MVC), the concentrations are different for different observations.

Regression models are typically applied to describe dependency between different numerical variables of one subject. In the case of homogeneous sample there exist many non-parametric estimators of the regression function, such as the Nadaraya-Watson estimator (NWE) and local linear regression estimator (LLRE). For homogeneous samples, NWE demonstrates an inappropriate bias in points where the regressor probability density function (PDF) has discontinuity (jump points). For such a scenario, the LLRE stands as a remedy, having a significantly smaller bias.

In this talk, we consider a modification of NWE (mNWE) and LLRE (mLLRE) for the estimation of the regression function of some MVC component. We will show that under suitable assumptions, the modified estimators are asymptotically normal. Moreover, the rate of convergence for the mNWE is different at different points of continuity and discontinuity of the regressor's PDF respectively, whereas the mLLRE preserves the same rate of convergence for both cases.

probabilitystatistics theorydata analysis, statistics and probability

Audience: researchers in the topic

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Asymptotic Methods in Statistics

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