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Adaptive and efficient estimation in the Gaussian sequence model

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  • Peng, Jingfu

Abstract

This paper deals with the problem of estimating the mean of an infinite-dimensional Gaussian vector by the principle of minimizing unbiased risk estimation (URE). The aim is to obtain an adaptive estimator to mimic the oracle with the smallest squared L2 loss/risk in a given class of linear estimators Λ. It is noticed that there is an essential balance between enlarging Λ for more efficient oracle risk and contracting Λ for the adaptivity to the oracle. This paper proves that this balance can be well achieved by minimizing URE in a list of intermediate classes between the projection and monotone classes. The resulting estimators are adaptive on the nonparametric space and have more efficient risks than the projection estimators under certain conditions.

Suggested Citation

  • Peng, Jingfu, 2023. "Adaptive and efficient estimation in the Gaussian sequence model," Statistics & Probability Letters, Elsevier, vol. 195(C).
    • Handle: RePEc:eee:stapro:v:195:y:2023:i:c:s0167715222002759
    DOI: 10.1016/j.spl.2022.109762
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    1. Gérard Kerkyacharian & Dominique Picard & Lucien Birgé & Peter Hall & Oleg Lepski & Enno Mammen & Alexandre Tsybakov & G. Kerkyacharian & D. Picard, 2000. "Thresholding algorithms, maxisets and well-concentrated bases," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 9(2), pages 283-344, December.
    2. Rivoirard, Vincent, 2004. "Maxisets for linear procedures," Statistics & Probability Letters, Elsevier, vol. 67(3), pages 267-275, April.
    3. Galtchouk, L. & Pergamenshchikov, S., 2006. "Asymptotically efficient estimates for nonparametric regression models," Statistics & Probability Letters, Elsevier, vol. 76(8), pages 852-860, April.
    4. Bruce E. Hansen, 2007. "Least Squares Model Averaging," Econometrica, Econometric Society, vol. 75(4), pages 1175-1189, July.
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