Estimation of a semiparametric mixture of regressions model

Abstract

We introduce in this paper a new mixture of regressions model which is a generalisation of the semiparametric two-component mixture model studied in Bordes, Delmas, and Vandekerkhove [(2006b), ‘Semiparametric Estimation of a Two-component Mixture Model When a Component is Known’, Scandinavian Journal of Statistics, 33, 733–752]. Namely, we consider a two-component mixture of regressions model in which one component is entirely known while the proportion, the slope, the intercept, and the error distribution of the other component are unknown. Our model is said to be semiparametric in the sense that the probability density function (pdf) of the error involved in the unknown regression model cannot be modelled adequately by using a parametric density family. When the pdfs of the errors involved in each regression model are supposed to be zero-symmetric, we propose an estimator of the various (Euclidean and functional) parameters of the model, and establish under mild conditions their almost sure rates of convergence. Finally, the implementation and numerical performances of our method are discussed using several simulated data sets and one real high-density array data set (ChIP-mix model).

Keywords:

• M-estimator
• mixture
• regression model
• empirical process
• semiparametric identifiability
• uniform convergence rate

AMS 2000 Subject Classifications :

• Primary: 62G05
• 62G20
• 62G08
• Secondary: 62E10

Acknowledgements

The author thanks the referees for their helpful and constructive comments. He is also very grateful to Philippe Barbe, David Hunter and Marie-Laure Martin Magniettes for their help and good advice during the writing of this manuscript.