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Sobolev-Hermite versus Sobolev nonparametric density estimation on $${\mathbb {R}}$$ R

Author

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  • Denis Belomestny

    (Duisburg-Essen University
    National Research University Higher School of Economics)

  • Fabienne Comte

    (Université Paris Descartes, Sorbonne Paris Cité)

  • Valentine Genon-Catalot

    (Université Paris Descartes, Sorbonne Paris Cité)

Abstract

In this paper, our aim is to revisit the nonparametric estimation of a square integrable density f on $${\mathbb {R}}$$ R , by using projection estimators on a Hermite basis. These estimators are studied from the point of view of their mean integrated squared error on $${\mathbb {R}}$$ R . A model selection method is described and proved to perform an automatic bias variance compromise. Then, we present another collection of estimators, of deconvolution type, for which we define another model selection strategy. Although the minimax asymptotic rates of these two types of estimators are mainly equivalent, the complexity of the Hermite estimators is usually much lower than the complexity of their deconvolution (or kernel) counterparts. These results are illustrated through a small simulation study.

Suggested Citation

  • Denis Belomestny & Fabienne Comte & Valentine Genon-Catalot, 2019. "Sobolev-Hermite versus Sobolev nonparametric density estimation on $${\mathbb {R}}$$ R," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 71(1), pages 29-62, February.
    • Handle: RePEc:spr:aistmt:v:71:y:2019:i:1:d:10.1007_s10463-017-0624-y
    DOI: 10.1007/s10463-017-0624-y
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    References listed on IDEAS

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    1. Gwennaëlle Mabon, 2017. "Adaptive Deconvolution on the Non-negative Real Line," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 44(3), pages 707-740, September.
    2. Chaleyat-Maurel, Mireille & Genon-Catalot, Valentine, 2006. "Computable infinite-dimensional filters with applications to discretized diffusion processes," Stochastic Processes and their Applications, Elsevier, vol. 116(10), pages 1447-1467, October.
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    Citations

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    Cited by:

    1. F. Comte & V. Genon-Catalot, 2020. "Regression function estimation on non compact support in an heteroscesdastic model," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 83(1), pages 93-128, January.
    2. F. Comte & V. Genon-Catalot, 2020. "Regression function estimation as a partly inverse problem," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 72(4), pages 1023-1054, August.
    3. Fabienne Comte & Nicolas Marie, 2021. "Nonparametric estimation for I.I.D. paths of fractional SDE," Statistical Inference for Stochastic Processes, Springer, vol. 24(3), pages 669-705, October.

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