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2020 Adaptive estimation in the supremum norm for semiparametric mixtures of regressions
Heiko Werner, Hajo Holzmann, Pierre Vandekerkhove
Electron. J. Statist. 14(1): 1816-1871 (2020). DOI: 10.1214/20-EJS1699


We investigate a flexible two-component semiparametric mixture of regressions model, in which one of the conditional component distributions of the response given the covariate is unknown but assumed symmetric about a location parameter, while the other is specified up to a scale parameter. The location and scale parameters together with the proportion are allowed to depend nonparametrically on covariates. After settling identifiability, we provide local M-estimators for these parameters which converge in the sup-norm at the optimal rates over Hölder-smoothness classes. We also introduce an adaptive version of the estimators based on the Lepski-method. Sup-norm bounds show that the local M-estimator properly estimates the functions globally, and are the first step in the construction of useful inferential tools such as confidence bands. In our analysis we develop general results about rates of convergence in the sup-norm as well as adaptive estimation of local M-estimators which might be of some independent interest, and which can also be applied in various other settings. We investigate the finite-sample behaviour of our method in a simulation study, and give an illustration to a real data set from bioinformatics.


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Heiko Werner. Hajo Holzmann. Pierre Vandekerkhove. "Adaptive estimation in the supremum norm for semiparametric mixtures of regressions." Electron. J. Statist. 14 (1) 1816 - 1871, 2020.


Received: 1 October 2019; Published: 2020
First available in Project Euclid: 24 April 2020

zbMATH: 07200245
MathSciNet: MR4090786
Digital Object Identifier: 10.1214/20-EJS1699

Keywords: adaptive estimation , M-estimation , semiparametric mixture , switching regression , uniform rates of convergence


Vol.14 • No. 1 • 2020
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