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Convergence rates in nonparametric estimation of level sets

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  • Baíllo, Amparo
  • Cuesta-Albertos, Juan A.
  • Cuevas, Antonio

Abstract

A level set of type {f[less-than-or-equals, slant]c} (where f is a density on and c is a positive value) can be estimated by its empirical version , where denotes a nonparametric (kernel) density estimator. We analyze, from two different points of view, the asymptotic behavior of the probability content of . Our results are motivated by applications in cluster analysis and outlier detection. Although the mathematical treatment is quite different in both cases, the conclusions are basically coincident. Roughly speaking, we show that the convergence rates are at most of type n-1/(d+2). For the univariate case d=1 this would be in the same spirit of the classical cube-root results found in some nonparametric setups.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:stapro:v:53:y:2001:i:1:p:27-35
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    References listed on IDEAS

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    1. Chanda, Kamal C. & Ruymgaart, F. H., 1989. "Asymptotic estimate of probability of misclassification for discriminant rules based on density estimates," Statistics & Probability Letters, Elsevier, vol. 8(1), pages 81-88, May.
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    Citations

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    Cited by:

    1. Cai, J., 2012. "Estimation concerning risk under extreme value conditions," Other publications TiSEM a92b089f-bc4c-41c2-b297-c, Tilburg University, School of Economics and Management.
    2. 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.
    3. Baíllo, Amparo, 2003. "Total error in a plug-in estimator of level sets," DES - Working Papers. Statistics and Econometrics. WS ws032806, Universidad Carlos III de Madrid. Departamento de Estadística.
    4. Delicado, Pedro & Vieu, Philippe, 2015. "Optimal level sets for bivariate density representation," Journal of Multivariate Analysis, Elsevier, vol. 140(C), pages 1-18.
    5. Cholaquidis, Alejandro & Fraiman, Ricardo & Moreno, Leonardo, 2022. "Level set and density estimation on manifolds," Journal of Multivariate Analysis, Elsevier, vol. 189(C).
    6. Cadre, BenoI^t, 2006. "Kernel estimation of density level sets," Journal of Multivariate Analysis, Elsevier, vol. 97(4), pages 999-1023, April.
    7. Ren, Qunshu & Mojirsheibani, Majid, 2008. "Nonparametric estimation of level sets under minimal assumptions," Statistics & Probability Letters, Elsevier, vol. 78(17), pages 3029-3033, December.
    8. Mammen, Enno & Polonik, Wolfgang, 2013. "Confidence regions for level sets," Journal of Multivariate Analysis, Elsevier, vol. 122(C), pages 202-214.
    9. Terán, Pedro & López-Díaz, Miguel, 2014. "Strong consistency and rates of convergence for a random estimator of a fuzzy set," Computational Statistics & Data Analysis, Elsevier, vol. 77(C), pages 130-145.
    10. 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.
    11. Baíllo, Amparo, 2003. "Total error in a plug-in estimator of level sets," Statistics & Probability Letters, Elsevier, vol. 65(4), pages 411-417, December.
    12. J Morio & R Pastel, 2012. "Plug-in estimation of d-dimensional density minimum volume set of a rare event in a complex system," Journal of Risk and Reliability, , vol. 226(3), pages 337-345, June.
    13. Elena Bernardino & Thomas Laloë & Rémi Servien, 2015. "Estimating covariate functions associated to multivariate risks: a level set approach," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 78(5), pages 497-526, July.
    14. Ghislaine Gayraud & Judith Rousseau, 2007. "Consistency results on nonparametric Bayesian estimation of level sets using spatial priors," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 16(1), pages 90-108, May.
    15. 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.

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