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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.
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
Full Text: DOI Euclid
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