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Kernel estimation of density level sets

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  • Cadre, BenoI^t

Abstract

Let f be a multivariate density and fn be a kernel estimate of f drawn from the n-sample X1,...,Xn of i.i.d. random variables with density f. We compute the asymptotic rate of convergence towards 0 of the volume of the symmetric difference between the t-level set {f[greater-or-equal, slanted]t} and its plug-in estimator {fn[greater-or-equal, slanted]t}. As a corollary, we obtain the exact rate of convergence of a plug-in-type estimate of the density level set corresponding to a fixed probability for the law induced by f.

Suggested Citation

  • Cadre, BenoI^t, 2006. "Kernel estimation of density level sets," Journal of Multivariate Analysis, Elsevier, vol. 97(4), pages 999-1023, April.
    • Handle: RePEc:eee:jmvana:v:97:y:2006:i:4:p:999-1023
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    References listed on IDEAS

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    1. Baíllo, Amparo & Cuesta-Albertos, Juan A. & Cuevas, Antonio, 2001. "Convergence rates in nonparametric estimation of level sets," Statistics & Probability Letters, Elsevier, vol. 53(1), pages 27-35, May.
    2. Cuevas, Antonio & Febrero, Manuel & Fraiman, Ricardo, 2001. "Cluster analysis: a further approach based on density estimation," Computational Statistics & Data Analysis, Elsevier, vol. 36(4), pages 441-459, June.
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    4. Nolan, D., 1991. "The excess-mass ellipsoid," Journal of Multivariate Analysis, Elsevier, vol. 39(2), pages 348-371, November.
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    2. Berthet, Philippe & Einmahl, John, 2020. "Cube Root Weak Convergence of Empirical Estimators of a Density Level Set," Other publications TiSEM 69103be2-c944-4ca1-b9e1-2, Tilburg University, School of Economics and Management.
    3. Christopher R. Genovese & Marco Perone-Pacifico & Isabella Verdinelli & Larry Wasserman, 2016. "Non-parametric inference for density modes," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 78(1), pages 99-126, January.
    4. Dau, Hai Dang & Laloë, Thomas & Servien, Rémi, 2020. "Exact asymptotic limit for kernel estimation of regression level sets," Statistics & Probability Letters, Elsevier, vol. 161(C).
    5. Mammen, Enno & Polonik, Wolfgang, 2013. "Confidence regions for level sets," Journal of Multivariate Analysis, Elsevier, vol. 122(C), pages 202-214.
    6. Leying Guan & Robert Tibshirani, 2022. "Prediction and outlier detection in classification problems," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 84(2), pages 524-546, April.
    7. King, Maxwell L. & Zhang, Xibin & Akram, Muhammad, 2020. "Hypothesis testing based on a vector of statistics," Journal of Econometrics, Elsevier, vol. 219(2), pages 425-455.
    8. Pedro Delicado & Philippe Vieu, 2017. "Choosing the most relevant level sets for depicting a sample of densities," Computational Statistics, Springer, vol. 32(3), pages 1083-1113, September.
    9. Biau, Gérard & Cadre, Benoît & Pelletier, Bruno, 2008. "Exact rates in density support estimation," Journal of Multivariate Analysis, Elsevier, vol. 99(10), pages 2185-2207, November.
    10. Lasse Holmström & Kyösti Karttunen & Jussi Klemelä, 2017. "Estimation of level set trees using adaptive partitions," Computational Statistics, Springer, vol. 32(3), pages 1139-1163, September.
    11. Chiwoo Park & Jianhua Z. Huang & Yu Ding, 2010. "A Computable Plug-In Estimator of Minimum Volume Sets for Novelty Detection," Operations Research, INFORMS, vol. 58(5), pages 1469-1480, October.
    12. Delicado, Pedro & Vieu, Philippe, 2015. "Optimal level sets for bivariate density representation," Journal of Multivariate Analysis, Elsevier, vol. 140(C), pages 1-18.
    13. Xiaochun Meng & James W. Taylor & Souhaib Ben Taieb & Siran Li, 2020. "Scores for Multivariate Distributions and Level Sets," Papers 2002.09578, arXiv.org, revised Jun 2023.
    14. Elena Di Bernardino & Thomas Laloë & Véronique Maume-Deschamps & Clémentine Prieur, 2013. "Plug-in estimation of level sets in a non-compact setting with applications in multivariate risk theory," Post-Print hal-00580624, HAL.
    15. Yen-Chi Chen & Christopher R. Genovese & Larry Wasserman, 2017. "Density Level Sets: Asymptotics, Inference, and Visualization," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 112(520), pages 1684-1696, October.

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