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Convergence Rates of Density Estimation in Besov Spaces

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DOI: 10.4236/am.2011.210175    3,819 Downloads   6,445 Views   Citations
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ABSTRACT

The optimality of a density estimation on Besov spaces Bsr,q(R) for the Lp risk was established by Donoho, Johnstone, Kerkyacharian and Picard (“Density estimation by wavelet thresholding,” The Annals of Statistics, Vol. 24, No. 2, 1996, pp. 508-539.). To show the lower bound of optimal rates of convergence Rn(Bsr,q, p), they use Korostelev and Assouad lemmas. However, the conditions of those two lemmas are difficult to be verified. This paper aims to give another proof for that bound by using Fano’s Lemma, which looks a little simpler. In addition, our method can be used in many other statistical models for lower bounds of estimations.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

H. Wang, "Convergence Rates of Density Estimation in Besov Spaces," Applied Mathematics, Vol. 2 No. 10, 2011, pp. 1258-1262. doi: 10.4236/am.2011.210175.

References

[1] G. Kerkyacharian and D. Picard, “Density Estimation in Besov Spaces,” Statistics & Probability Letters, Vol. 13, No. 1, 1992, pp. 15-24. doi:10.1016/0167-7152(92)90231-S
[2] D. L. Donoho, I. M. Johnstone, G. Kerkyacharian and D. Picard, “Den-sity Estimation by Wavelet Thresholding,” The Annals of Statistics, Vol. 24, No. 2, 1996, pp. 508-539. doi:10.1214/aos/1032894451
[3] W. H?rdle, G. Ker-kyacharian, D. Picard and A. B. Tsybakov, “Wavelets, Approximation and Statistical Applications,” Sprin-ger-Verlag, New York, 1997.
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[5] P. Baldi, G. Ker-kyacharian, D. Marinucci and D. Picard, “Adaptive Den-sity Estimation for Directional Data Using Needlets,” The Annals of Statistics, Vol. 37, No. 6A, 2009, pp. 3362-3395. doi:10.1214/09-AOS682
[6] C. Christophe, “Regression with Random Design: A Minimax Study,” Statistics & Probability Letters, Vol. 77, No. 1, 2007, pp. 40-53. doi:10.1016/j.spl.2006.05.010
[7] A. B. Tsyba-kov, “Optimal Rates of Aggregation,” COLT/Kernel 2003 Lecture Notes in Artificial Intelligence 2777, Springer, Heidelberg, 2003, pp. 303-313.

  
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