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Density estimation by kernel and wavelets methods: Optimality of Besov spaces

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  • Kerkyacharian, Gérard
  • Picard, Dominique

Abstract

This paper is showing that the saturation space of the minimax rate associated to a Lp loss and linear estimators is the Besov space Bs[infinity]p. More precisely, it is shown that if a function space included in Lp is such that its minimax rate is the usual one s/(1 + 2s) and if this rate is attained by a sequence of linear estimators, then this space is included in a ball of the space Bs[infinity]p. This implies, for example, that the minimax rates that have been estimated for the Sobolev balls are in fact only a consequence of their inclusions in such Besov balls

Suggested Citation

  • Kerkyacharian, Gérard & Picard, Dominique, 1993. "Density estimation by kernel and wavelets methods: Optimality of Besov spaces," Statistics & Probability Letters, Elsevier, vol. 18(4), pages 327-336, November.
    • Handle: RePEc:eee:stapro:v:18:y:1993:i:4:p:327-336
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    Cited by:

    1. Hohage, Thorsten & Maréchal, Pierre & Simar, Léopold & Vanhems, Anne, 2024. "A Mollifier Approach To The Deconvolution Of Probability Densities," Econometric Theory, Cambridge University Press, vol. 40(2), pages 320-359, April.
    2. Marina Vannucci & Brani Vidakovic, 1997. "Preventing the Dirac disaster: Wavelet based density estimation," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 6(2), pages 145-159, August.
    3. Kato, Takeshi, 1999. "Density estimation by truncated wavelet expansion," Statistics & Probability Letters, Elsevier, vol. 43(2), pages 159-168, June.
    4. Gaëlle Chagny & Claire Lacour, 2015. "Optimal adaptive estimation of the relative density," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 24(3), pages 605-631, September.
    5. Gérard, Kerkyacharian & Dominique, Picard, 1997. "Limit of the quadratic risk in density estimation using linear methods," Statistics & Probability Letters, Elsevier, vol. 31(4), pages 299-312, February.
    6. Karine Bertin & Vincent Rivoirard, 2009. "Maxiset in sup-norm for kernel estimators," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 18(3), pages 475-496, November.
    7. Rivoirard, Vincent, 2004. "Maxisets for linear procedures," Statistics & Probability Letters, Elsevier, vol. 67(3), pages 267-275, April.
    8. Gérard Kerkyacharian & Dominique Picard & Lucien Birgé & Peter Hall & Oleg Lepski & Enno Mammen & Alexandre Tsybakov & G. Kerkyacharian & D. Picard, 2000. "Thresholding algorithms, maxisets and well-concentrated bases," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 9(2), pages 283-344, December.
    9. Hoffmann, Marc, 1999. "Adaptive estimation in diffusion processes," Stochastic Processes and their Applications, Elsevier, vol. 79(1), pages 135-163, January.
    10. Hoffmann, Marc, 1997. "Minimax estimation of the diffusion coefficient through irregular samplings," Statistics & Probability Letters, Elsevier, vol. 32(1), pages 11-24, February.
    11. Durastanti, Claudio, 2016. "Adaptive global thresholding on the sphere," Journal of Multivariate Analysis, Elsevier, vol. 151(C), pages 110-132.
    12. Liang Han-Ying & Mammitzsch Volker & Steinebach Josef, 2005. "Nonlinear wavelet density and hazard rate estimation for censored data under dependent observations," Statistics & Risk Modeling, De Gruyter, vol. 23(3), pages 161-180, March.
    13. Pinheiro, Aluisio & Vidakovic, Brani, 1997. "Estimating the square root of a density via compactly supported wavelets," Computational Statistics & Data Analysis, Elsevier, vol. 25(4), pages 399-415, September.
    14. Aya-Moreno, Carlos & Geenens, Gery & Penev, Spiridon, 2018. "Shape-preserving wavelet-based multivariate density estimation," Journal of Multivariate Analysis, Elsevier, vol. 168(C), pages 30-47.
    15. Koo, Ja-Yong & Kim, Woo-Chul, 1996. "Wavelet density estimation by approximation of log-densities," Statistics & Probability Letters, Elsevier, vol. 26(3), pages 271-278, February.
    16. Liang, Han-Ying & de Uña-Álvarez, Jacobo, 2011. "Wavelet estimation of conditional density with truncated, censored and dependent data," Journal of Multivariate Analysis, Elsevier, vol. 102(3), pages 448-467, March.

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