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Laguerre and Hermite bases for inverse problems

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Abstract

We present inverse problems of nonparametric statistics which have a smart solution using projection estimators on bases of functions with non compact support, namely, a Laguerre basis or a Hermite basis. The models are Yi = XiUi, Zi = Xi + Σi, where the Xi’s are i.i.d. with unknown density f, the Σi’s are i.i.d. with known density fΣ, the Uii’s are i.i.d. with uniform density on [0, 1]. The sequences (Xi), (Ui), (x03A3;i) are independent. We define projection estimators off in the two cases of indirect observations of (X1,..., Xn), and we give upper bounds for their L2-risks on specific Sobolev-Laguerre or Sobolev-Hermite spaces. Data-driven procedures are described and proved to perform automatically the bias-variance compromise.

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Authors and Affiliations

  1. MAP5, UMR CNRS 8145, Université Paris Descartes, France

    F. Comte & V. Genon-Catalot

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  1. F. Comte
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  2. V. Genon-Catalot
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Correspondence to F. Comte.

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Comte, F., Genon-Catalot, V. Laguerre and Hermite bases for inverse problems. J. Korean Stat. Soc. 47, 273–296 (2018). https://doi.org/10.1016/j.jkss.2018.03.001

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