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Density Level Sets: Asymptotics, Inference, and Visualization

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  • Yen-Chi Chen
  • Christopher R. Genovese
  • Larry Wasserman

Abstract

We study the plug-in estimator for density level sets under Hausdorff loss. We derive asymptotic theory for this estimator, and based on this theory, we develop two bootstrap confidence regions for level sets. We introduce a new technique for visualizing density level sets, even in multidimensions, which is easy to interpret and efficient to compute. Supplementary materials for this article are available online.

Suggested Citation

  • 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.
    • Handle: RePEc:taf:jnlasa:v:112:y:2017:i:520:p:1684-1696
    DOI: 10.1080/01621459.2016.1228536
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    References listed on IDEAS

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    1. Cadre, BenoI^t, 2006. "Kernel estimation of density level sets," Journal of Multivariate Analysis, Elsevier, vol. 97(4), pages 999-1023, April.
    2. Ilya S. Molchanov, 1998. "A Limit Theorem for Solutions of Inequalities," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 25(1), pages 235-242, March.
    3. Mammen, Enno & Polonik, Wolfgang, 2013. "Confidence regions for level sets," Journal of Multivariate Analysis, Elsevier, vol. 122(C), pages 202-214.
    4. J. Kruskal, 1964. "Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis," Psychometrika, Springer;The Psychometric Society, vol. 29(1), pages 1-27, March.
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    Cited by:

    1. Alberto Muñoz & Gabriel Martos & Javier Gonzalez, 2023. "Level Sets Semimetrics for Probability Measures with Applications in Hypothesis Testing," Methodology and Computing in Applied Probability, Springer, vol. 25(1), pages 1-17, March.
    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. 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).
    4. Merlo, Luca & Petrella, Lea & Salvati, Nicola & Tzavidis, Nikos, 2022. "Marginal M-quantile regression for multivariate dependent data," Computational Statistics & Data Analysis, Elsevier, vol. 173(C).
    5. Paula Saavedra-Nieves & Rosa M. Crujeiras, 2022. "Nonparametric estimation of directional highest density regions," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 16(3), pages 761-796, September.
    6. Cholaquidis, Alejandro & Fraiman, Ricardo & Moreno, Leonardo, 2022. "Level set and density estimation on manifolds," Journal of Multivariate Analysis, Elsevier, vol. 189(C).
    7. Aneiros, Germán & Horová, Ivana & Hušková, Marie & Vieu, Philippe, 2022. "On functional data analysis and related topics," Journal of Multivariate Analysis, Elsevier, vol. 189(C).
    8. 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.

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