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Estimator Selection: a New Method with Applications to Kernel Density Estimation

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  • Claire Lacour

    (University Paris-Sud)

  • Pascal Massart

    (University Paris-Sud)

  • Vincent Rivoirard

    (Université Paris Dauphine)

Abstract

Estimator selection has become a crucial issue in non parametric estimation. Two widely used methods are penalized empirical risk minimization (such as penalized log-likelihood estimation) or pairwise comparison (such as Lepski’s method). Our aim in this paper is twofold. First we explain some general ideas about the calibration issue of estimator selection methods. We review some known results, putting the emphasis on the concept of minimal penalty which is helpful to design data-driven selection criteria. Secondly we present a new method for bandwidth selection within the framework of kernel density density estimation which is in some sense intermediate between these two main methods mentioned above. We provide some theoretical results which lead to some fully data-driven selection strategy.

Suggested Citation

  • Claire Lacour & Pascal Massart & Vincent Rivoirard, 2017. "Estimator Selection: a New Method with Applications to Kernel Density Estimation," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 79(2), pages 298-335, August.
    • Handle: RePEc:spr:sankha:v:79:y:2017:i:2:d:10.1007_s13171-017-0107-5
    DOI: 10.1007/s13171-017-0107-5
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    References listed on IDEAS

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    1. Lacour, C. & Massart, P., 2016. "Minimal penalty for Goldenshluger–Lepski method," Stochastic Processes and their Applications, Elsevier, vol. 126(12), pages 3774-3789.
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    Cited by:

    1. Van Ha Hoang & Thanh Mai Pham Ngoc & Vincent Rivoirard & Viet Chi Tran, 2022. "Nonparametric estimation of the fragmentation kernel based on a partial differential equation stationary distribution approximation," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 49(1), pages 4-43, March.
    2. Saif Eddin Jabari & Nikolaos M. Freris & Deepthi Mary Dilip, 2020. "Sparse Travel Time Estimation from Streaming Data," Transportation Science, INFORMS, vol. 54(1), pages 1-20, January.
    3. Comte, Fabienne & Marie, Nicolas, 2023. "Nonparametric drift estimation from diffusions with correlated Brownian motions," Journal of Multivariate Analysis, Elsevier, vol. 198(C).
    4. Pham Ngoc, Thanh Mai, 2019. "Adaptive optimal kernel density estimation for directional data," Journal of Multivariate Analysis, Elsevier, vol. 173(C), pages 248-267.
    5. Marie, Nicolas, 2022. "Projection estimators of the stationary density of a differential equation driven by the fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 180(C).
    6. Florian Dussap, 2023. "Nonparametric multiple regression by projection on non-compactly supported bases," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 75(5), pages 731-771, October.
    7. Dimitris N. Politis & Peter F. Tarassenko & Vyacheslav A. Vasiliev, 2022. "Estimating Smoothness and Optimal Bandwidth for Probability Density Functions," Stats, MDPI, vol. 6(1), pages 1-20, December.

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