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August
2020, 19(8): 4007-4022.
doi: 10.3934/cpaa.2020177
Uncompactly supported density estimation with $ L^{1} $ risk
| 1. | Department of Applied Mathematics, Beijing University of Technology, Pingle Yuan 100, Beijing, 100124, China |
| 2. | School of Mathematics and Information Science, Weifang University, Weifang, Shandong, 261061, China |
The perfect achievements have been made for $ L^{p}\; (1\leq p<+\infty) $ risk estimation, when a density function has compact support. However, there does not exist $ L^{1} $ risk estimation for uncompactly supported densities in general. Motivated by the work of Juditsky & Lambert-Lacroix (A. Juditsky and S. Lambert-Lacroix, On minimax density estimation on $ \mathbb{R} $, Bernoulli, 10(2004), 187-220) and Goldenshluger & Lepski (A. Goldenshluger and O. Lepski, On adaptive minimax density estimation on $ \mathbb{R}^{d} $, Probab. Theory Relat. Fields., 159(2014), 479-543), we provide an adaptive estimate for a family of density functions not necessarily having compact supports in this paper.
Mathematics Subject Classification:
Primary: 62G07, 42C40, 62G20.
Citation:
Kaikai Cao, Youming Liu. Uncompactly supported density estimation with $ L^{1} $ risk. Communications on Pure and Applied Analysis,
2020, 19
(8)
: 4007-4022.
doi: 10.3934/cpaa.2020177
![]()
References:
| [1] |
L. Birgé,
On estimating a density using hellinger distance and some other strange facts, Probab. Theory Relat. Fields, 71 (1986), 271-291.
doi: 10.1007/BF00332312. |
| [2] |
L. Birgé,
Model selection for density estimation with $L^2$-loss, Probab. Theory Relat. Fields, 158 (2014), 533-574.
doi: 10.1007/s00440-013-0488-x. |
| [3] |
J. Bretagnolle and C. Huber,
Estimation des densites: risque minimax, Z Wahrscheinlichkeitstheorie Verw Geb., 47 (1979), 119-137.
doi: 10.1007/BF00535278. |
| [4] |
K. K. Cao and Y. M. Liu, On the Reynaud-Bouret–Rivoiard–Tuleau-Malot problem, Int. J. Wavelets Multiresolut. Inform. Process., 16 (2018), 1850038.
doi: 10.1142/S0219691318500388. |
| [5] |
L. Devroye and L. Györfi, Nonparametric Density Estimation: The $L^{1}$ View, Wiley, New York, 1985. |
| [6] |
D. L. Donoho, I. M. Johnstone, G. Kerkyacharian and D. Picard,
Density estimation by wavelet thresholding, Ann. Statist., 24 (1996), 508-539.
doi: 10.1214/aos/1032894451. |
| [7] |
E. Giné and R. Nickl, Mathematical foundations of infinite-dimensional statistics model, Cambridge university Press, Cambridge, 2015.
doi: 10.1017/CBO9781107337862.
|
| [8] |
A. Goldenshluger and O. Lepski,
On adaptive minimax density estimation on $\mathbb{R}^{d}$, Probab. Theory Relat Fields, 159 (2014), 479-543.
doi: 10.1007/s00440-013-0512-1. |
| [9] |
W. Härdle, G. Kerkyacharian, D. Picard and A. Tsybakov., Wavelets, Approximation and Statistical Applications, Springer-Verlag, New York, 1998.
doi: 10.1007/978-1-4612-2222-4. |
| [10] |
A. Juditsky and S. Lambert-Lacroix,
On minimax density estimation on $\mathbb{R}$, Bernoulli, 10 (2004), 187-220.
doi: 10.3150/bj/1082380217. |
| [11] |
G. Kerkyacharian and D. Picard,
Density estimation in Besov space, Statist. Probab. Lett., 13 (1992), 15-24.
doi: 10.1016/0167-7152(92)90231-S. |
| [12] |
O. Lepski,
Multivariate estimation under sup-norm loss: oracle approach, adaption and independence structure, Ann. Statist., 41 (2013), 1005-1034.
doi: 10.1214/13-AOS1109. |
| [13] |
P. Reynaud-Bouret, V. Rivoirard and C. Tuleau-Malot,
Adaptive density estimation: a curse of support?, J. Statist. Plan. Infer., 141 (2011), 115-139.
doi: 10.1016/j.jspi.2010.05.017. |
show all references
References:
| [1] |
L. Birgé,
On estimating a density using hellinger distance and some other strange facts, Probab. Theory Relat. Fields, 71 (1986), 271-291.
doi: 10.1007/BF00332312. |
| [2] |
L. Birgé,
Model selection for density estimation with $L^2$-loss, Probab. Theory Relat. Fields, 158 (2014), 533-574.
doi: 10.1007/s00440-013-0488-x. |
| [3] |
J. Bretagnolle and C. Huber,
Estimation des densites: risque minimax, Z Wahrscheinlichkeitstheorie Verw Geb., 47 (1979), 119-137.
doi: 10.1007/BF00535278. |
| [4] |
K. K. Cao and Y. M. Liu, On the Reynaud-Bouret–Rivoiard–Tuleau-Malot problem, Int. J. Wavelets Multiresolut. Inform. Process., 16 (2018), 1850038.
doi: 10.1142/S0219691318500388. |
| [5] |
L. Devroye and L. Györfi, Nonparametric Density Estimation: The $L^{1}$ View, Wiley, New York, 1985. |
| [6] |
D. L. Donoho, I. M. Johnstone, G. Kerkyacharian and D. Picard,
Density estimation by wavelet thresholding, Ann. Statist., 24 (1996), 508-539.
doi: 10.1214/aos/1032894451. |
| [7] |
E. Giné and R. Nickl, Mathematical foundations of infinite-dimensional statistics model, Cambridge university Press, Cambridge, 2015.
doi: 10.1017/CBO9781107337862.
|
| [8] |
A. Goldenshluger and O. Lepski,
On adaptive minimax density estimation on $\mathbb{R}^{d}$, Probab. Theory Relat Fields, 159 (2014), 479-543.
doi: 10.1007/s00440-013-0512-1. |
| [9] |
W. Härdle, G. Kerkyacharian, D. Picard and A. Tsybakov., Wavelets, Approximation and Statistical Applications, Springer-Verlag, New York, 1998.
doi: 10.1007/978-1-4612-2222-4. |
| [10] |
A. Juditsky and S. Lambert-Lacroix,
On minimax density estimation on $\mathbb{R}$, Bernoulli, 10 (2004), 187-220.
doi: 10.3150/bj/1082380217. |
| [11] |
G. Kerkyacharian and D. Picard,
Density estimation in Besov space, Statist. Probab. Lett., 13 (1992), 15-24.
doi: 10.1016/0167-7152(92)90231-S. |
| [12] |
O. Lepski,
Multivariate estimation under sup-norm loss: oracle approach, adaption and independence structure, Ann. Statist., 41 (2013), 1005-1034.
doi: 10.1214/13-AOS1109. |
| [13] |
P. Reynaud-Bouret, V. Rivoirard and C. Tuleau-Malot,
Adaptive density estimation: a curse of support?, J. Statist. Plan. Infer., 141 (2011), 115-139.
doi: 10.1016/j.jspi.2010.05.017. |
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