Estimation of a semiparametric mixture of regressions model

Abstract

We consider in this paper a contamined regression model where the distribution of the contaminating component is known when the Euclidean parameters of the regression model, the noise distribution, the contamination ratio and the distribution of the design data are unknown. Our model is said to be semiparametric in the sense that the probability density function (pdf) of the noise involved in the regression model is not supposed to belong to a parametric density family. When the pdf's of the noise and the contaminating phenomenon are supposed to be symmetric about zero, we propose an estimator of the various (Euclidean and functionnal) parameters of the model, and prove under mild conditions its convergence. We prove in particular that, under technical conditions all satisfied in the Gaussian case, the Euclidean part of the model is estimated at the rate o a.s (n −1/4+γ ), γ > 0. We recall that, as it is pointed out in Bordes and Vandekerkhove [5], this result cannot be ignored to go further in the asymptotic theory for this class of models. Finally the implementation and numerical performances of our method are discussed on several toy examples.

Laplace mixture of linear experts

“…t and skew-t distributions), which may be more suitable in different contexts; these generalizations constitute an interesting direction for future work. Furthermore, it would be interesting to investigate the combination of the MoLE framework for incorporating covariate dependencies with the semiparametric mixtures of linear regression models of Hunter and Young (2012) and Vandekerkhove (2013). We hope that this article provokes further research into the application of LMoLE models and the extension of the MoLE framework.…”

An Overview of Semiparametric Extensions of Finite Mixture Models

“…For example, Vandekerkhove (2013) studied a two-component mixture of regressions model in which one component is entirely known while the mixing proportion, the slope, the intercept, and the error distribution of the other component are unknown. The method proposed by Vandekerkhove (2013) performs well for data sets of reasonable size, but since it is based on the optimization of a contrast function of size O(n 2 ), the performance is not desirable as the sample size increases. Bordes et al (2013) also studied the same model as Vandekerkhove (2013), and proposed a new method-of-moments estimator, whose order is of O(n).…”

“…The method proposed by Vandekerkhove (2013) performs well for data sets of reasonable size, but since it is based on the optimization of a contrast function of size O(n 2 ), the performance is not desirable as the sample size increases. Bordes et al (2013) also studied the same model as Vandekerkhove (2013), and proposed a new method-of-moments estimator, whose order is of O(n). Young (2014) extended the mixture of linear regression models to incorporate changepoints, by assuming one or more of the components are piecewise linear.…”

Semiparametric estimation of a two-component mixture of linear regressions in which one component is known

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A new estimation method for the two-component mixture model introduced in [28] is proposed. This model consists of a two-component mixture of linear regressions in which one component is entirely known while the proportion, the slope, the intercept and the error distribution of the other component are unknown.

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