Wavelet estimations for densities and their derivatives with Fourier oscillating noises
© Guo and Liu; licensee Springer. 2014
Received: 24 March 2014
Accepted: 28 May 2014
Published: 8 June 2014
By developing the classical kernel method, Delaigle and Meister provide a nice estimation for a density function with some Fourier-oscillating noises over a Sobolev ball and over risk (Delaigle and Meister in Stat. Sin. 21:1065-1092, 2011). The current paper extends their theorem to Besov ball and risk with by using wavelet methods. We firstly show a linear wavelet estimation for densities in over risk, motivated by the work of Delaigle and Meister. Our result reduces to their theorem, when . Because the linear wavelet estimator is not adaptive, a nonlinear wavelet estimator is then provided. It turns out that the convergence rate is better than the linear one for . In addition, our conclusions contain estimations for density derivatives as well.
1 Introduction and preliminary
We assume that all and are independent and the density function of the noise δ is known.
Here and (non-negative integer set). Such noises are called Fourier-oscillating. Clearly, (2) allows having zeros for . When , (2) reduces to (1) (non-zero case).
under the assumption (2) (Theorem 4.1 in ). Here, stands for the Sobolev ball with the radius L. This above convergence rate attains the same one as in the non-zero case [1–4, 6]. In particular, it does not depend on the parameter v.
It seems that many papers deal with estimations. However, estimations () are important [5, 12]. On the other hand, Besov spaces contain many classical spaces (e.g., Sobolev spaces and Hölder spaces) as special examples. The current paper extends (3) from to the Besov ball , and from to risk estimations. In addition, our results contain estimations for d th derivatives of . The next section provides a linear wavelet estimation for over a Besov ball and over risk (). It turns out that our estimation reduces to (3), when , . Moreover, we show a nonlinear wavelet estimation which improves the linear one for in the last part.
1.1 Wavelet basis
, (the integer set);
if and only if for each ;
(the space is dense in );
There exists (scaling function) such that forms an orthonormal basis of .
where , .
A family of important examples are Daubechies wavelets , which are compactly supported in time domain . They can be smooth enough with increasing supports as N gets large, although do not have analytic formulas except for .
By using Proposition 8.3 in , we have the following conclusion.
1.2 Besov spaces
One of the advantages of wavelet bases is that they can characterize Besov spaces. To introduce those spaces (see ), we need the Sobolev spaces with integer order and . Then can be considered as .
stands for the smoothness modulus of f. It should be pointed out that .
According to Theorem 9.6 in , the following result holds.
, where is the projection operator to .
Here and throughout, denotes for some constant ; means ; we use standing for both and .
2 Linear wavelet estimation
We shall provide a linear wavelet estimation for a compactly supported density function and its derivatives under Fourier-oscillating noises in this section, motivated by the work of Delaigle and Meister. It turns out that our result generalizes their theorem.
where J and depend only on v and the support length of .
Then it can easily be seen that , and .
in this paper.
Finally, . This completes the proof of the first part of Lemma 2.1. □
Now, we are in a position to state our first theorem.
Proof It is easy to see that for , because is continuously embedded into . Moreover, thanks to Jensen’s inequality.
Take . Then the inequality (14) follows, and the proof of Theorem 2.1 is finished. □
Remark 2.1 If and , then , , Theorem 2.1 reduces to Theorem 4.1 in .
Remark 2.2 From the choice in the proof of Theorem 2.1, we find that our estimator is not adaptive, because it depends on the parameter s of . In order to avoid that shortcoming, we study a nonlinear estimation in the next part.
3 Nonlinear estimation
where , and . Clearly, the cardinality since both and ψ have compact supports.
with . This completes the proof of Lemma 2.1. □
In the proof of (26), one needs to choose .
In this above proof, one needs to choose .
where as defined in (19).
Since for , and . Then one obtains the desired inequality (28).
for , one finds by direct computations. Hence, in each case, which is (33). Now, (24) follows from (26), (27), (29), and (33). The proof is done. □
Remark 3.1 We find easily from Theorem 2.1 and Theorem 3.1 that the nonlinear wavelet estimator converges faster than the linear one for . Moreover, the nonlinear estimator is adaptive, while the linear one is not.
Remark 3.2 This paper studies wavelet estimations of a density and its derivatives with Fourier-oscillating noises. The remaining problems include the optimality of the above estimations, numerical experiments as well as the corresponding regression problems. We shall investigate those problems in the future.
This paper is supported by the National Natural Science Foundation of China (No. 11271038).
- Walter GG: Density estimation in the presence of noise. Stat. Probab. Lett. 1999, 41: 237–246. 10.1016/S0167-7152(98)00160-6View ArticleMATHGoogle Scholar
- Pensky M, Vidakovic B: Adaptive wavelet estimator for nonparametric density deconvolution. Ann. Stat. 1999, 27: 2033–2053. 10.1214/aos/1017939249MathSciNetView ArticleMATHGoogle Scholar
- Fan J: On the optimal rates of convergence for nonparametric deconvolution problems. Ann. Stat. 1991, 19: 1257–1272. 10.1214/aos/1176348248View ArticleMATHGoogle Scholar
- Fan J, Koo J: Wavelet deconvolution. IEEE Trans. Inf. Theory 2002, 48: 734–747. 10.1109/18.986021MathSciNetView ArticleMATHGoogle Scholar
- Lounici K, Nickl R: Global uniform risk bounds for wavelet deconvolution estimators. Ann. Stat. 2011, 39: 201–231. 10.1214/10-AOS836MathSciNetView ArticleMATHGoogle Scholar
- Li R, Liu Y: Wavelet optimal estimations for a density with some additive noises. Appl. Comput. Harmon. Anal. 2014, 36: 416–433. 10.1016/j.acha.2013.07.002MathSciNetView ArticleMATHGoogle Scholar
- Sun, J, Morrison, H, Harding, P, Woodroofe, M: Density and mixture estimation from data with measurement errors. Technical report (2002)Google Scholar
- Devroye L: Consistent deconvolution in density estimation. Can. J. Stat. 1989, 17: 235–239. 10.2307/3314852MathSciNetView ArticleMATHGoogle Scholar
- Hall P, Meister A: A ridge-parameter approach to deconvolution. Ann. Stat. 2007, 35: 1535–1558. 10.1214/009053607000000028MathSciNetView ArticleMATHGoogle Scholar
- Meister A: Deconvolution from Fourier-oscillating error densities under decay and smoothness restrictions. Inverse Probl. 2008, 24: 1–14.MathSciNetView ArticleGoogle Scholar
- Delaigle A, Meister A: Nonparametric function estimation under Fourier-oscillating noise. Stat. Sin. 2011, 21: 1065–1092. 10.5705/ss.2009.082MathSciNetView ArticleMATHGoogle Scholar
- Donoho DL, Johnstone IM, Kerkyacharian G, Picard D: Density estimation by wavelet thresholding. Ann. Stat. 1996, 24: 508–539.MathSciNetView ArticleMATHGoogle Scholar
- Hernández E, Weiss G: A First Course on Wavelets. CRC Press, Boca Raton; 1996.View ArticleMATHGoogle Scholar
- Daubechies I: Ten Lectures on Wavelets. SIAM, Philadelphia; 1992.View ArticleMATHGoogle Scholar
- Härdle W, Kerkyacharian G, Picard D, Tsybakov A Lecture Notes in Statist. In Wavelets, Approximation, and Statistical Applications. Springer, New York; 1998.View ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.