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Exact rates in density support estimation

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  • Biau, Gérard
  • Cadre, Benoît
  • Pelletier, Bruno

Abstract

Let f be an unknown multivariate probability density with compact support Sf. Given n independent observations X1,...,Xn drawn from f, this paper is devoted to the study of the estimator of Sf defined as unions of balls centered at the Xi and of common radius rn. To measure the proximity between and Sf, we employ a general criterion dg, based on some function g, which encompasses many statistical situations of interest. Under mild assumptions on the sequence (rn) and some analytic conditions on f and g, the exact rates of convergence of are obtained using tools from Riemannian geometry. The conditions on the radius sequence are found to be sharp and consequences of the results are discussed from a statistical perspective.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:jmvana:v:99:y:2008:i:10:p:2185-2207
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    References listed on IDEAS

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    1. Tsybakov, A.B. & Korostelev, A.P. & Simar, L., 1992. "Efficient Estimation of Monotone Boundaries," Papers 9209, Catholique de Louvain - Institut de statistique.
    2. Hardle, Wolfgang & Tsybakov, A. B., 1993. "How sensitive are average derivatives?," Journal of Econometrics, Elsevier, vol. 58(1-2), pages 31-48, July.
    3. Klemelä, Jussi, 2004. "Complexity penalized support estimation," Journal of Multivariate Analysis, Elsevier, vol. 88(2), pages 274-297, February.
    4. Korostelev, A. P. & Simar, L. & Tsybakov, A. B., 1995. "Estimation of monotone boundaries," LIDAM Reprints CORE 1178, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    5. Hardle, W. & Park, B. U. & Tsybakov, A. B., 1995. "Estimation of Non-sharp Support Boundaries," Journal of Multivariate Analysis, Elsevier, vol. 55(2), pages 205-218, November.
    6. Cadre, BenoI^t, 2006. "Kernel estimation of density level sets," Journal of Multivariate Analysis, Elsevier, vol. 97(4), pages 999-1023, April.
    7. Hardle, Wolfgang & Tsybakov, A. B., 1993. "How sensitive are average derivatives?," Journal of Econometrics, Elsevier, vol. 58(1-2), pages 31-48, July.
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    Cited by:

    1. Mammen, Enno & Polonik, Wolfgang, 2013. "Confidence regions for level sets," Journal of Multivariate Analysis, Elsevier, vol. 122(C), pages 202-214.

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