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Adaptive estimation in the supremum norm for semiparametric mixtures of regressions. (English) Zbl 1440.62279
A two-component semiparametric mixture of regressions model is studied $Y=W(\mu(X)+\epsilon_1)+(1-W)\sigma(X)\epsilon_2,$ where a random vector $$X$$ has a compact support. The explanatory variable $$X$$ and the response variable $$Y$$ are observable while the latent variable $$W$$ and error variables $$\epsilon_1$$ and $$\epsilon_2$$ are not. The unknown location and scaling functions $$\mu(\cdot)$$ and $$\sigma(\cdot)$$ partially determine the distributional relationship between $$X$$ and $$Y$$ along with the unknown mixing function $$p(\cdot).$$ Conditionally on $$\{X=x\},$$ the variable $$W$$ has a Bernoulli distribution with parameter $$p(x).$$ Conditionally on $$\{X=x\},$$ the vectors $$\epsilon_1$$ and $$\epsilon_2$$ have zero-symmetric conditional pdfs $$f_x$$ and $$\bar{f},$$ respectively, where $$\bar{f}$$ is assumed known and not to depend on $$x$$, while $$f_x$$ is unknown and may depend on $$x.$$ The functional parameter $$\theta(x)=(p(x), \sigma(x), \mu(x), f_x)$$ collects all the $$x$$-local quantities to be estimated from the data.
Based on $$n$$ independent copies of the model, local M-estimator of $$\theta(\cdot)$$ is constructed which converges in the sup-norm at the optimal rates over Hölder-smoothness classes. An adaptive version of the estimator is introduced based on the method in [O. V. Lepskij, Theory Probab. Appl. 36, No. 4, 682–697 (1991; Zbl 0776.62039); translation from Teor. Veroyatn. Primen. 36, No. 4, 645–659 (1991)]. Simulations investigate the finite-sample behavior of the method, and an illustration is given to a real data set from bioinformatics.
##### MSC:
 62J05 Linear regression; mixed models 62H30 Classification and discrimination; cluster analysis (statistical aspects) 62P10 Applications of statistics to biology and medical sciences; meta analysis
Zbl 0776.62039
ChIPmix
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