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Robust estimation of a regression function in exponential families

3 November 2020
Y. Baraud
Juntong Chen
ArXiv (abs)PDFHTML
Abstract

We observe nnn pairs of independent random variables X1=(W1,Y1),…,Xn=(Wn,Yn)X_{1}=(W_{1},Y_{1}),\ldots,X_{n}=(W_{n},Y_{n})X1​=(W1​,Y1​),…,Xn​=(Wn​,Yn​) and assume, although this might not be true, that for each i∈{1,…,n}i\in\{1,\ldots,n\}i∈{1,…,n}, the conditional distribution of YiY_{i}Yi​ given WiW_{i}Wi​ belongs to a given exponential family with real parameter θi⋆=θ⋆(Wi)\theta_{i}^{\star}=\boldsymbol{\theta}^{\star}(W_{i})θi⋆​=θ⋆(Wi​) the value of which is an unknown function θ⋆\boldsymbol{\theta}^{\star}θ⋆ of the covariate WiW_{i}Wi​. Given a model Θ‾\boldsymbol{\overline\Theta}Θ for θ⋆\boldsymbol{\theta}^{\star}θ⋆, we propose an estimator θ^\boldsymbol{\widehat \theta}θ with values in Θ‾\boldsymbol{\overline\Theta}Θ the construction of which is independent of the distribution of the WiW_{i}Wi​. We show that θ^\boldsymbol{\widehat \theta}θ possesses the properties of being robust to contamination, outliers and model misspecification. We establish non-asymptotic exponential inequalities for the upper deviations of a Hellinger-type distance between the true distribution of the data and the estimated one based on θ^\boldsymbol{\widehat \theta}θ. We deduce a uniform risk bound for θ^\boldsymbol{\widehat \theta}θ over the class of H\"olderian functions and we prove the optimality of this bound up to a logarithmic factor. Finally, we provide an algorithm for calculating θ^\boldsymbol{\widehat \theta}θ when θ⋆\boldsymbol{\theta}^{\star}θ⋆ is assumed to belong to functional classes of low or medium dimensions (in a suitable sense) and, on a simulation study, we compare the performance of θ^\boldsymbol{\widehat \theta}θ to that of the MLE and median-based estimators. The proof of our main result relies on an upper bound, with explicit numerical constants, on the expectation of the supremum of an empirical process over a VC-subgraph class. This bound can be of independent interest.

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