Convergence Rates of Density Estimation in Besov Spaces ()
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.
Share and Cite:
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.
Conflicts of Interest
The authors declare no conflicts of interest.
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.
|
|
[4]
|
A. B. Tsybakov, “In-troduction to Nonparametric Estimation,” (English) Revised and Extended from the 2004 French Original, Translated by Vladimir Zaiats, Springer Series in Statistics, Springer, New York, 2009.
|
|
[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.
|