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

A new estimation method for the two-component mixture model introduced in Vandekerkhove (2012) is proposed. This model, which 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, seems to be of interest for the analysis of large datasets produced from two-color ChIPchip high-density microarrays. In spite of good performance for datasets of reasonable size, the method proposed in Vandekerkhove (2012) suffers from a serious drawback when the sample size becomes large, as it is based on the optimization of a contrast function whose pointwise computation requires O(n2) operations. The range of applicability of the method derived in this work is substantially larger as it is based on a method-of-moment estimator whose computation only requires O(n) operations. From a theoretical perspective, the asymptotic normality of both the estimator of the Euclidean parameter vector and of the semiparametric estimator of the c.d.f. of the error is proved under weak conditions not involving the zero-symmetry assumption typically used this last decade. The finite-sample performance of the latter estimators is studied