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

* Corresponding author

Received  June 2019 Revised  January 2020 Published  May 2020

Fund Project: Supported by National Natural Science Foundation of China Grant 11771030, and the Science and Technology Program of Beijing Municipal Commission of Education Grant KM202010005025

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.

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. DonohoI. M. JohnstoneG. 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-BouretV. 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. DonohoI. M. JohnstoneG. 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-BouretV. 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.

Figure 1.  Graph of $ g_l $
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