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Inverse statistical learning. (English) Zbl 1349.62102
Summary: Let $$(X,Y)\in\mathcal{X}\times\mathcal{Y}$$ be a random couple with unknown distribution $$P$$. Let $$\mathcal{G}$$ be a class of measurable functions and $$\ell$$ a loss function. The problem of statistical learning deals with the estimation of the Bayes: $g^{*}=\operatorname{arg} \underset{g\in \mathcal{G}}{\min} \mathbb{E}_{P} \ell(g,(X,Y)).$ In this paper, we study this problem when we deal with a contaminated sample $$(Z_{1},Y_{1}),\dots,(Z_{n},Y_{n})$$ of i.i.d. indirect observations. Each input $$Z_{i}$$, $$i=1,\dots,n$$ is distributed from a density $$Af$$, where $$A$$ is a known compact linear operator and $$f$$ is the density of the direct input $$X$$.
We derive fast rates of convergence for the excess risk of empirical risk minimizers based on regularization methods, such as deconvolution kernel density estimators or spectral cut-off. These results are comparable to the existing fast rates in [V. Koltchinskii, Ann. Stat. 34, No. 6, 2593–2706 (2006; Zbl 1118.62065)] for the direct case. It gives some insights into the effect of indirect measurements in the presence of fast rates of convergence.

##### MSC:
 62G05 Nonparametric estimation 62H30 Classification and discrimination; cluster analysis (statistical aspects)
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