# Wavelet Optimal Estimations for Density Functions under Severely Ill-Posed Noises

Authors: Rui Li; Youming Liu
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

Motivated by Lounici and Nickl's work (2011), this paper considers the problem of estimation of a densityf based on an independent and identically distributed sample Y1,…,Yn
from g=f*φ. We show a wavelet optimal estimation for a density (function) over Besov ball Br,qs(L) and Lp risk (1≤p<∞) in the presence of severely ill-posed noises. A wavelet linear estimation is firstly presented. Then, we prove a lower bound, which shows our wavelet estimator optimal. In other words, nonlinear wavelet estimations are not needed in that case. It turns out that our results extend some theorems of Pensky and Vidakovic (1999), as well as Fan and Koo (2002).


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## Body

## 1. Introduction and Preliminary

Wavelets have made great achievements in studying the statistical modelY=X+ϵ, where X stands for real-valued random variable with unknown probability density f, and ϵ denotes an independent random noise (error) with density φ.In 1999, Pensky and Vidakovic [1] investigate Meyer wavelet estimation over Sobolev spaces W2s(ℝ) and L2 risk under moderately and severely ill-posed noises. Three years later, Fan and Koo [2] extend those works from W2s(ℝ) to Besov spaces Br,qs(ℝ)(1≤r≤2). It should be pointed out that, by using different method, Lounici and Nickl [3] study wavelet optimal estimation over Besov spaces B∞,∞s(ℝ) and L∞ risk under both noises. In [4], we provide a wavelet optimal estimation over Br,qs(ℝ) and Lp risk (1≤p<∞,r,q∈[1,∞]) under moderately ill-posed noise. This current paper deals with the same problem under the severely ill-posed noises. It turns out that our result contains some theorems of [1, 2] as special cases. Our discussion also shows that nonlinear wavelet estimations are not needed for severely ill-posed noise, which is totally different with moderately ill-posed case.Letϕ and ψ∈L2(ℝ) be a scaling and mother wavelet function, respectively. Then each f∈L2(ℝ) has an expansion (in L2 sense):
(1)f=∑k∈ℤ‍αJkϕJk+∑j≥J‍∑k∈ℤ‍βjkψjk,
with αjk:=〈f,ϕjk〉 and βjk:=〈f,ψjk〉. Here and throughout, we use the standard notation hjk(x):=2j/2h(2jx-k) in wavelet analysis [5]. A class of important wavelets are Meyer's, whose Fourier transforms are C∞ and compactly supported on {t:2π/3≤|t|≤8π/3} [5]. It is easy to see that ∀a≥0, ∃Ca>0 such that |x|a|ϕ(x)|≤Ca. In this paper, the Fourier transform f^ for f∈L(ℝ) is defined by
(2)f^(t):=(Ff)(t)=∫ℝ‍f(x)e-itxdx.
The classical method extends that definition to L2(ℝ) functions.The following two lemmas are fundamental in our discussions. We use∥f∥p to denote Lp(ℝ) norm of f∈Lp(ℝ), and ∥λ∥p do lp(ℤ) norm of λ∈lp(ℤ), where
(3)lp(ℤ):={{λ={λk},∑k∈ℤ‍|λk|p<∞},1≤p<∞,{λ={λk},supk∈ℤ|λk|<∞},p=∞.Lemma 1 (see [6]).
Leth be a scaling or a wavelet function with supx∈ℝ∑k∈ℤ|h(x-k)|<∞. Then, there exist C2≥C1>0 such that for λ={λk}∈lp(ℤ) with 1≤p≤∞,
(4)C12j(1/2-1/p)∥λ∥p≤∥∑k∈ℤ‍λkhjk∥p≤C22j(1/2-1/p)∥λ∥p.One of the advantages of wavelet bases is that they can characterize Besov spaces. To introduce those spaces [6], we need the well-known Sobolev spaces with integer exponents
(5)Wpm(ℝ):={f∈Lp(ℝ),f(m)∈Lp(ℝ)},
with the Sobolev norm ∥f∥Wpm:=∥f∥p+∥f(m)∥p. Then Lp(ℝ) can be considered as Wp(0)(ℝ). For 1≤p, q≤∞ and s=m+α with α∈(0,1], a Besov space is defined by
(6)Br,qs(ℝ):={f∈Wrm(ℝ),∥t-αωr2(f(m),t)∥q*<∞},
with the norm ∥f∥srq:=∥f∥Wrm+∥t-αωr2(f(m),t)∥q*, where ωr2(f,t):=sup|h|≤t∥f(·+2h)-2f(·+h)+f(·)∥r denotes the smoothness modulus of f and
(7)∥h∥q*={(∫0∞(|h(t)|qt)dt)1/q,if1≤q<∞,esssupt∈ℝ|h(t)|,ifq=∞.Lemma 2 (see [6]).
Letϕ be a Meyer scaling function and ψ be the corresponding wavelet. If f∈Lr(ℝ), 1≤r≤∞, α0k=∫‍f(x)ϕ0k(x)dx, and βjk=∫‍f(x)ψjk(x)dx, then the following assertions are equivalent:(i)
f∈Br,qs(ℝ),1≤q≤∞;(ii)
{2js∥Pjf-f∥r}j≥0∈lq, where Pjf:=∑k∈ℤαjkϕjk;(iii)
∥α0k∥r+∥{2j(s+1/2-1/r)∥βj·∥r}j≥0∥q<∞.In each case,(8)∥f∥srq~∥α0·∥r+∥{2j(s+1/2-1/r)∥βj·∥r}j≥0∥q.
Here and after, A≲B denotes A≤CB for some constant C>0; A≳B means B≲A; A~B stands for both A≲B and B≲A, α0· does for the sequence of {α0k,k∈ℤ}.At the end of this subsection, we make some assumptions on noise densityφ, which will be dealt with in this current paper. For α>0, c>0, β∈ℝ,(C1)
|φ^(t)|≤C(1+|t|2)-β/2e-c|t|α(a.e.);(C2)
|φ^′(t)|≤C(1+|t|2)-β/2e-c|t|α(a.e.);(C3)
|φ^(t)|≥C(1+|t|2)-β/2e-c|t|α(a.e.).Clearly, the classical Cauchy densities satisfy (C1)–(C3) with α=c=1 and β=0, and the Gaussian density does satisfy (C1)–(C3) with α=2, c=1/2, and β=0. It should be pointed out that those above conditions (C1)–(C3) are a little different with [2].In the next section, we define a wavelet linear estimator and provide an upper bound estimation over Besov spacesBr,qs(ℝ) and Lp risk under the condition (C3); the third part gives a lower bound estimation which shows the result of Section 2 optimal; some concluding remarks are discussed in the last part.


## 2. Upper Bound

To introduce the main theorem of this section, we assume thatY1,Y2,…,Yn are independent and identically distributed (i.i.d) random variables of Y=X+ϵ, the density φ of random noise ϵ satisfies condition (C3), and ϕ stands for the Meyer scaling function. As in [1], define
(9)(Kjϕ)(y):=12π∫ℝ‍eityϕ^(t)φ^(-2jt)dt,α^jk:=1n∑i=1n‍(Kjϕ)jk(Yi),
as well as a linear wavelet estimator
(10)fnL:=∑|k|≤Kn‍α^jkϕjk,
(the positive integer Kn will be given later on). Then α^j·∈l∞, fnL is well defined, and EfnL=∑|k|≤Knαjkϕjk.We use suppf to stand for the support of f and |suppf| to do its length. Moreover, for L>0, denote
(11)Br,qs(L):={f∈Br,qs(ℝ),f≥0,∫ℝ‍f(x)dx=1and∥f∥srq≤L},Br,qs(L,M)={f∈Br,qs(L),|suppf|≤MforsomeM>0}.
It is reasonable to assume L>1 for r=1, since ∥f∥s1q≥∥f∥1=1 in that case.Theorem 3.
Letφ satisfy (C3) and ϕ be the Meyer scaling function. If p∈[1,∞), q,r∈[1,∞], ∥x2f∥μ≤A(μ≥1,A>0), then, with Kn~e(lnn)θ(0<θ<1), (3/8π)((lnn)/4c)1/α < 2j≤(3/4π)((lnn)/4c)1/α, s′:=s-(1/r-1/p)+, and x+=max{x,0},
(12)supf∈Br,qs(L,M)E∥fnL-f∥pp≲(lnn)-(s′/α)p.
In particular, f∈Br,qs(L,M) can be replaced by f∈Br,qs(L), when r≤p.Proof.
Whenr≤p, s′:=s-(1/r-1/p)+=s-1/r+1/p. Since lr is continuously embedded to lp, Lemma 2 implies Br,qs(ℝ)⊆Bp,qs′(ℝ). Hence,
(13)supf∈Br,qs(L)E∥fnL-f∥pp≲supf∈Bp,qs′(L)E∥fnL-f∥pp.
When r>p, one obtains that, for some C>0, Br,qs(L,M)⊆Bp,qs(CL,M) and
(14)supf∈Br,qs(L,M)E∥fnL-f∥pp≲supf∈Bp,qs(CL,M)E∥fnL-f∥pp.
In fact, f∈Br,qs(L,M) and Hölder inequality imply that ∥f∥pp=∫‍|f(x)|pdx≲∥f(x)∥rp due to |suppf|≤M. By the definition of Besov norm, ∥f∥spq≤C∥f∥srq. According to (13) and (14), it is sufficient to prove
(15)supf∈Bp,qs(L)E∥fnL-f∥pp≲(lnn)-(s/α)p,
for the conclusion of Theorem 3.
Recall thatfnL=∑|k|≤Knα^jkϕjk and EfnL=∑|k|≤Knαjkϕjk. Then
(16)E∥fnL-f∥pp≲E∥fnL-EfnL∥pp+∥∑|k|>Kn‍αjkϕjk∥pp+∥Pjf-f∥pp.
By f∈Bp,qs(L) and Lemma 2,
(17)∥Pjf-f∥p≲2-js.
To estimate the middle term of (16), one observes that αjk=∫ℝ‍ϕjk(x)f(x)dx, |k2αjk| ≤ ∫ℝ‍|2jx-k-2jx|2|ϕjk(x)|f(x)dx≲∫ℝ‍|2jx-k|2|ϕjk(x)|f(x)dx+∫ℝ‍|2jx|2|ϕjk(x)|f(x)dx. Since ϕ is the Meyer scaling function, supx∈ℝ|x|2|ϕ(x)|<∞ and
(18)∫ℝ‍|2jx-k|2|ϕjk(x)|f(x)dx≤2j/2supx∈ℝ|x|2|ϕ(x)|.
On the other hand, ∫ℝ‍|2jx|2|ϕjk(x)|f(x)dx≤22j∥x2f∥μ×∥ϕjk∥μ′≲2j(5/2-1/μ′)∥x2f∥μ with 1/μ + 1/μ′=1. Therefore |k2αjk|≲2j/2+2j(5/2-1/μ′)≲2j(5/2) and ∑|k|>Kn|αjk|p ≲ ∑|k|>Knk-2p2j(5/2)p ≲ Kn1-2p2j(5/2)p. This with Lemma 1 leads to
(19)∥∑|k|>Kn‍αjkϕjk∥pp≲2j(p/2-1)∑|k|>Kn‍|αjk|p≤2j(3p-1)Kn1-2p.
Now, it remains to considerE∥fnL-EfnL∥pp: Using fnL-EfnL=∑|k|≤Kn(α^jk-αjk)ϕjk and Lemma 1, one knows
(20)E∥fnL-EfnL∥pp≲2j(p/2-1)∑|k|≤Kn‍E|α^jk-αjk|p.
Clearly, α^jk-αjk=(1/n)∑i=1n[(Kjϕ)jk(Yi)-E(Kjϕ)jk(Yi)]. Define Xi,k:=(Kjϕ)jk(Yi)-E(Kjϕ)jk(Yi). Then EXi,k=0 and α^jk-αjk=(1/n)∑i=1nXi,k. To apply Rosenthal's inequality (Proposition 10.2, [6]), one estimates |Xi,k| and E|Xi,k|p: note that |(Kjϕ)jk(Yi)|=2j/2|(1/2π) × ∫ℝ‍eit(2jYi-k)(ϕ^(t)/φ^(-2jt))dt|≲2j/2∫ℝ‍(1+|2jt|2)β/2ec|2jt|α×|ϕ^(t)|dt≲2j(β+1/2)ec((4π/3)2j)α due to (C3) and suppϕ^⊆[-(4/3)π,(4/3)π]. Then
(21)|Xi,k|≲2j(β+1/2)ec((4π/3)2j)α,E|Xi,k|p≲2j(β+1/2)pecp((4π/3)2j)α.
BecauseXi,k are i.i.d, the Rosenthal's inequality tells that
(22)E|α^jk-αjk|p≲{n-p[nE|Xi,k|p+np/2(E|Xi,k|2)p/2],p>2,n-p/2(E|Xi,k|2)p/2,0<p≤2.
This with (21) implies that, for p≥2, E|α^jk-αjk|p≲n1-p2j(β+1/2)pecp((4π/3)2j)αI{p > 2} + n-p/22j(β+1/2)p×ecp((4π/3)2j)α≲n-p/22j(β+1/2)pecp((4π/3)2j)α. Moveover, (20) reduces to
(23)E∥fnL-EfnL∥pp≲Knn-p/22j(β+1)p-jecp((4π/3)2j)α.
Then it follows from (16)–(19) and (23) that
(24)E∥fnL-f∥pp≲2-jps+2j(3p-1)Kn1-2p+Knn-p/22j(β+1)p-jecp((4π/3)2j)α.
By the choices Kn~e(lnn)θ and (3/8π)((lnn)/4c)1/α<2j≤(3/4π)((lnn)/4c)1/α (stated in Theorem 3), one receives that 2-jps≲(lnn)-(s/α)p,
(25)2j(3p-1)Kn1-2p≲(lnn)(3p-1)/αe-(2p-1)(lnn)θ=o((lnn)-(s/α)p),Knn-p/22j(β+1)p-jecp((4π/3)2j)α≲e(lnn)θn-p/2(lnn)((β+1)p-1)/αnp/4=o((lnn)-(s/α)p).
Finally, the desired conclusion (15) follows.Remark 4.
Note that the choices ofj and Kn do not depend on the unknown parameters s, r, and q. Then our linear wavelet estimator fnL over Besov space Br,qs is adaptive or implementable. The same conclusion holds for L∞ and L2 estimations; see Theorem 2 in [3] and Corollary 1 in [1]. On the other hand, when p=2 and 1≤r≤2, our Theorem 3 reduces to Theorem 4 in [2]; from the proof of Theorem 3, we find that, for p>1, the assumption ∥x2f(x)∥μ≤A can be replaced by ∥xf(x)∥∞≤A, which is the same as in [1]. Therefore, for p=r=q=2, Theorem 3 of [1] follows directly from our Theorem 3.


## 3. Lower Bound

In this part, we will provide a lower bound estimation, which shows Theorem3 to be the best possible in some sense. The following lemmas are needed in the proof of our main theorem of this section.Lemma 5.
Lethη(x):=ηp(ηx) with p(x)=1/π(1+x2), η>0, and r,q≥1. Then for L>0 (L>2 when r=1), there exists η0>0 such that h0:=hη0∈Br,qs(L/2). If ψ is the Meyer wavelet function and |λk|≤d2-j/2(k=1,2,…,2j), then, for some small d>0,
(26)|∑k=12j‍λkψjk(x)|≤h0(x).Proof.
It is easy to see that(1+x2)-1∈Wrm(ℝ) (for r≥1) and hη∈Br,qs(ℝ)(s<m) by the definition of Besov space. Since
(27)∥hη∥srq=η1-1/r∥p∥r+η1+⌊s⌋-1/r∥p(⌊s⌋)∥r+η1+s-1/r∥ωr2(p(⌊s⌋),t)ts-⌊s⌋∥q*,
where ⌊s⌋ denotes the largest integer no more than s,∥h0∥srq can be made small enough by choosing small η0>0, when r>1. Clearly, L>2 is needed, when r=1.
If|λk|≤d2-j/2(k=1,2,…,2j), then |∑k=12j‍λkψjk(x)|≤d∑k=12j‍|ψ(2jx-k)|≲d∑k=12j(1+|2jx-k|2)-1 because ψ is the Meyer function. Note that ∑k=12j‍(1+|2jx-k|2)-1=∑k=12j‍(1+22j|x - 2-jk|2)-1 ≤ 2-2j∑k=12j‍(|x|2-1)-1=2-j(x2-1)-1. Then for some small d>0 and |x|≥2,
(28)|∑k=12j‍λkψjk(x)|≤2-jh0(x).
Hence, (26) holds for |x|≥2. On the other hand, when |x|<2, h0(x)≳1 and |∑k=12j‍λkψjk(x)| ≤ d∑k=12j‍|ψ(2jx-k)|≤dsupx∑k∈ℤ‍|ψ(x-k)|≤Cd≤h0(x). Therefore, (26) is true, when d>0 small enough. This completes the proof of Lemma 5.The next lemma extends an estimate in the proof of Theorem 1 in [3].Lemma 6.
Letψ be the Meyer wavelet function, h0(x), defined as in Lemma 5. If φ satisfies (C1), (C2), and ωk∈{0,1}, then
(29)∫ℝ‍(h0*φ)-1(y)(∑k=12j‍ωkψjk*φ)2(y)dy≲2-2jβe-2c((2π/3)2j)α∑k=12j‍ωk2.Proof.
As shown in proof of Theorem 1 of [3], one finds easily that (h0*φ)(y)≳(1+y2)-1 and therefore
(30)∫ℝ‍(h0*φ)-1(y)(∑k=12j‍ωkψjk*φ)2(y)dy≲∫ℝ‍(1+y2)(∑k=12j‍ωkψjk*φ)2(y)dy.
By Parseval identity, (C1) and suppψjk^⊆{y,|y|≥(2π/3)2j}, ∫ℝ‍(∑k=12jωkψjk*φ)2(y)dy = (1/2π)∫|t|≥(2π/3)2j‍×|∑k=12jωkψjk^(t)|2|φ^(t)|2dt≲2-2βje-2c((2π/3)2j)α∫|t|≥(2π/3)2j‍×|∑k=12jωkψjk^(t)|2dt. Moreover, the orthonormality of ψjk^ concludes that
(31)∫ℝ‍(∑k=12j‍ωkψjk*φ)2(y)dy≲C2-2βje-2c((2π/3)2j)α∑k=12j‍ωk2.
To estimate∫ℝ‍y2(∑k=12j‍ωkψjk*φ)2(y)dy, one proves an inequality:
(32)I:=∫ℝ‍|∑k=12j‍ωkddtψjk^(t)|2dt≤C∑k=12j‍ωk2.
Note that ∑k=12j‍ωkψ0k(x),  x∑k=12j‍ωkψ0k(x)∈L(ℝ), and ∑k=12j‍ωk(d/dt)ψjk^(t) = 2-(3/2)j[(d/dt)F(∑k=12j‍ωkψ0k)](2-jt). Then ∫ℝ‍[(d/dt)F(∑k=12j‍ωkψ0k)]2(t)dt=∫ℝ‍|ix|2|∑k=12j‍ωk × ψ0k(x)|2dx=∑k=12j‍∑k′=12j‍ωkωk′∫ℝ‍(x + k)2ψ(x)ψ[x-(k′-k)]dx and
(33)I=2-2j∑k=12j‍∑k′=12j‍ωkωk′∫ℝ‍(x+k)2ψ(x)ψ[x-(k′-k)]dx.
Since 〈ψ(x),ψ(x-k)〉=δk,0, ∑k=12j‍∑k′=12j‍ωkωk′k2∫ℝ‍ψ(x)ψ(x+k-k′)dx=∑k=12j‍ωk2k2≤22j∑k=12j‍ωk2; On the other hand, the boundedness of ∫ℝ‍x2ψ(x)ψ(x+l)dx and ∫ℝ‍xψ(x)ψ(x+l)dx implies that
(34)|∑k=12j‍∑k′=12j‍ωkωk′∫ℝ‍x2ψ(x)ψ(x+k-k′)dx|≲∑k=12j‍∑k′=12j‍ωkωk′≲2j∑k=12j‍ωk2,
as well as |∑k=12j‍∑k′=12j‍ωkωk′k∫ℝ‍xψ(x)ψ(x+k-k′)dx|≲22j∑k=12j‍ωk2. Hence, I≲∑k=12j‍ωk2, which reaches (32).
Defineq(t):=F(∑k=12jωkψjk*φ)(t). Then q,q′∈L(ℝ) and q is locally absolutely continuous. Therefore, q′^(y)=iyq^(y)=iy(∑k=12j‍ωkψjk*φ)(-y) and
(35)∫ℝ‍y2(∑k=12j‍ωkψjk*φ)2(y)dy=∫ℝ‍|q′^(y)|2dy=2π∫ℝ‍|q′(t)|2dt.
Clearly, q′(t)=∑k=12jωk[ψjk^(t)φ^′(t)+φ^(t)(d/dt)ψjk^(t)] and ∫ℝ‍|q′(t)|2dt ≤ 2[∫ℝ‍|φ^(t)∑k=12jωk(d/dt)ψjk^(t)|2dt+∫ℝ‍|φ^′(t)×∑k=12jωkψjk^(t)|2dt]≲2-2jβe-2c((2π/3)2j)α[∫ℝ‍|∑k=12jωk(d/dt)×ψjk^(t)|2dt+2∫ℝ‍|∑k=12jωkψjk^(t)|2dt]  thanks to (C1), (C2), and
(36)suppψjk^⊆{y,|y|≥2π32j}.
Moreover, ∫ℝ‍|q′(t)|2dt≤C2-2jβe-2c((2π/3)2j)α∑k=12jωk2 because of (32) and the orthonormality of ψjk. This with (35), (31), and (30) leads to the desired conclusion of Lemma 6.
Two more classical theorems play important roles in our discussions. We list the first one as Lemma7, which can be found in [7].Lemma 7 (Varshamov-Gilbert).
LetΩ={ω=(ω1,…,ωm),ωk∈{0,1}} with m≥8. Then there exists a subset {ω(0),…,ω(M)} of Ω such that M≥2m/8, ω(0)=(0,…,0), and for j,l=0,1,…,M,j≠l,
(37)∑k=1m‍|ωk(j)-ωk(l)|≥m8.Given two probability measuresP and Q on a measurable space (𝕏,ℱ), the Kullback divergence of P and Q is defined by
(38)K(P,Q):={∫ln(dPdQ)dP,ifP≪Q,+∞,otherwise.
Here, P≪Q stands for P absolutely continuous with respect to Q. In that case, K(P,Q)=∫ln(fP(x)/fQ(x))fP(x)dx, where the function fP(x) denotes the density function of P. The second classical theorem is taken from [8].Lemma 8 (Fano).
Let(𝕏,ℱ,Pk) be probability measurable spaces and Ak∈ℱ,  k=0,1,…,m. If Ak∩Av=∅ for k≠v, then
(39)sup0≤k≤mPk(Akc)≥min{12,C0me-Km},
where Km:=inf0≤v≤m(1/m)∑k≠vK(Pk,Pv), C0=e-3e-1, and Ac denotes the complement of a set A.Now, we are in the position to state the main theorem in this section.Theorem 9.
Letφ satisfy (C1) and (C2), and let fn(·):=fn(Y1,Y2,…,Yn,·) be an estimator of f∈Br,qs(L). Then for s>0, p∈[1,∞), q,r∈[1,∞], and s≥1/r, there exists C>0 independent of fn such that with s′:=s-(1/r-1/p)+,
(40)supf∈Br,qs(L)E∥fn-f∥pp≥C(lnn)-(s′/α)p.Proof.
Assume thatψ is the Meyer wavelet function, then ψjk∈Br,qs(ℝ). By Lemma 2,
(41)∥2-j(s+1/2)∑k=12j‍ωkψjk∥Br,qs≲2-j(s+1/2)2j(s-1/r+1/2)×(∑k=12j‍|ωk|r)1/r≤1,
for ωk∈{0,1}. Furthermore, with the function h0 defined in Lemma 5, there exists c1>0 such that hω:=h0+c12-j(s+1/2)∑k=12j‍ωkψjk≥0 and ∥hω∥srq≤L due to that Lemma. Define
(42)Λ:={hω=h0+c12-j(s+1/2)∑k=12j‍ωkψjk,ωk∈{0,1}}.
Then ∫‍hω(x)dx=1 because ∫‍ψ(x)dx=0 and ∫‍h0(x)dx=1.
By Lemma7, one finds Λ′:={hω(0),hω(1),…,hω(M)}⊆Λ with M≥22j/8 and hω(0)=h0 such that for ω≠ω′ and hω,hω′∈Λ′, ∑k=12j‍|ωk-ωk′|≥2j/8. It is easy to see that
(43)∥hω-hω′∥p=c12-j(s+1/2)∥∑k=12j‍(ωk-ωk′)ψjk∥p.
This with Lemma 1 leads to ∥hω-hω′∥p≳c12-j(s+1/2)×2j(1/2-1/p)(∑k=12j‍|ωk-ωk′|)1/p≳c12-js, and therefore
(44)θj:=minhω≠hω′∈Λ′∥hω-hω′∥p≳c12-js.
Define Aω:={∥fn-hω∥p<θj/2} for hω∈Λ′. Then Aω∩Aω′=∅, when ω≠ω′. Clearly, hω*φ is a density function because both hω and φ are density functions. Let Phω*φn be the probability measure on the Lebesgue space (ℝn,ℒ) with the density ∏i=1n‍(hω*φ)(yi). Then Lemma 8 tells that
(45)suphω∈Λ′Phω*φn{∥fn-hω∥p>θj2}≥min{12,C0M1/2e-KM}.
According to Lemma5, hω(x)≲h0(x) and Phω*φn≪Ph0*φn. Moreover, K(Phω*φn,Ph0*φn) = ∫ℝ⋉‍[ln(Πi=1n(hω*φ)(yi)/Πi=1n(h0*φ)(yi))]Πi=1n(hω*φ)(yi)dy1dy2⋯dyn=n∫ℝ‍ln((hω*φ)(y)/(h0*φ)(y))(hω*φ)(y)dy=n∫ℝ‍ln[1+c12-j(s+1/2)(∑k=12jωkψjk*φ)(h0*φ)-1](hω*φ)(y)dy. Since hω*φ/h0 * φ>0, c12-j(s+1/2)(∑k=12jωkψjk*φ)(h0*φ)-1=(hω*φ-h0*φ)/h0*φ>-1. Combining this with ln(1+x)≤x(x>-1), one knows
(46)K(Phω*φn,Ph0*φn)≤nc12-j(s+1/2)×∫ℝ‍(h0*φ)-1(y)×(∑k=12j‍ωkψjk*φ)(y)×(hω*φ)(y)dy.
Because ∫ℝ‍(ψjk*φ)(y)dy=F(ψjk*φ)(0)=ψjk^(0)φ^(0)=0, the above inequality reduces to K(Phω*φn,Ph0*φn)≤c12n2-2j(s+1/2)∫‍[(∑k=12jωkψjk*φ)2(y)][(h0*φ)(y)]-1dy≲c12n2-2j(s+1/2+β)exp{-2c((2π/3)2j)α}∑k=12jωk2≤c12n2-2j(s+β)×exp{-2c((2π/3)2j)α} thanks to Lemma 6. Hence,
(47)KM:=inf0≤v≤m1m∑k≠v‍K(Pk,Pv)≲c12n2-2j(s+β)exp{-2c(2π32j)α}.
Note thatM≥22j/8 and take j such that
(48)c-1(2π3)-αlnn<2jα≤2αc-1(2π3)-αlnn.
Then M1/2e-KM≥exp{((ln2)/16)(3/2π)c-1/α(lnn)1/α}exp{-Cc12n-1[c-1/α(3/2π)(lnn)1/α]-2(s+β)}≳1 (choose c1>0 small enough). Furthermore, (45) reduces to
(49)suphω∈Λ′Phω*φn{∥fn-hω∥p>θj2}≳1.
Hence, supf∈Br,qs(L)E∥fn-f∥pp≥suphω∈Λ′E∥fn-hω∥pp≥suphω∈Λ′Phω*φn{∥fn-hω∥p>θj/2}(θj/2)p≳θjp. This with (44) and (48) leads to
(50)supf∈Br,qs(L)E∥fn-f∥pp≳(lnn)-sp/α,
which is the desired conclusion of Theorem 9, when r≥p (s′=s in that case).
Whenr<p, s′=s-(1/r-1/p)+=s-1/r+1/p, it remains to show
(51)supf∈Br,qs(L)E∥fn-f∥pp≥C(lnn)-((s-1/r+1/p)/α)p.
Similar to the proof of (50), one takes small c2>0 such that
(52)hk(x):=h0(x)+c22-j(s-1/r+1/2)ψjk(x)(k=1,…,2j)
satisfies hk(x)≥0, hk∈Br,qs(L) and ∫‍hk(x)dx=1. Clearly, ∥hk-h0∥p=c22-j(s-1/r+1/2)2j(1/2-1/p)∥ψ∥p and ∥hk-hk′∥p=c22-j(s-1/r+1/p)∥ψ(·)-ψ(·-(k′-k))∥p for 1≤k≠k′≤2j. Since ψ is the Meyer wavelet function, infk≠0∥ψ(·)-ψ(·-k)∥p>0 and
(53)δj:=min0≤k≠k′≤2j∥hk-hk′∥p≳c22-j(s-1/r+1/p).
DefineAk:={∥fn-hk∥p<δj/2}(k=0,1,…,2j). Then Ak∩Av=∅(k≠v) and
(54)supk=0,1,…,2jPhk*φn{∥fn-hk∥p≥δj2}≥min{12,C02j/2e-K2j},
due to Lemma 8. Similar (even simpler) arguments to the estimation of KM show K2j ≲ c22n2-2j(s-1/r+β+1/2)e-2c((2π/3)2j)α. Taking j as in (48), one receives that
(55)K2j≤Cc22n-1[c-1(2π3)-αlnn]-(2(s-1/r+β+1/2))/α,
and 2j/2e-K2j≥[c-1(2π/3)-αlnn]1/2αexp{-Cc22[c-1(2π/3)-αlnn]-(2(s-1/r+β+1/2))/αn-1}≥1 by choosing small c2>0. Thus (54) reduces to
(56)supk=0,1,…,2jPhk*φn{∥fn-hk∥p≥δj2}≳1.
Moreover, supf∈Br,qs(L)Ef*φn∥fn-f∥pp≥supk=0,1,…,2jEhk*φn∥fn-hk∥pp ≥ supk=0,1,…,2jPhk*φn{∥fn-hk∥p ≥ δj/2}(δj/2)p≳δjp. This with (53) and (48) leads to (51). This completes the proof of Theorem 9.Remark 10.
By Theorems9 and 3, the linear wavelet estimator fnL is optimal for a density in Besov spaces with severely ill-posed noise. Therefore, we do not need to consider nonlinear wavelet estimations in that case. This contrasts sharply with moderately ill-posed noise case under which nonlinear wavelet estimation improves the linear one [2, 4].Remark 11.
Whenp=2 and 1≤r≤2, our Theorem 9 is better than Theorem 6 in [2], because (lnn)-s′/α>(lnn)-s/α. Moreover, Theorems 9 and 3 lead to Theorem 3 in that paper for p=2 and 1≤r≤2. In addition, our conditions (C1) and (C2) are a little weaker than the assumptions in [2].


## 4. Concluding Remarks

This paper provides anLp(1≤p<∞) risk upper bound for a linear wavelet estimator fnL (Theorem 3), which turns out to be optimal (Theorem 9). Therefore, nonlinear estimations are not needed under severely ill-posed noises. Although we assume p<∞ in Theorem 9, the proof of that theorem shows that, for p=∞,
(57)supf∈Br,qs(L)E∥fn-f∥∞≥C(lnn)-(s-1/r)/α.
In particular, when r=q=∞, this above estimation reduces to partial result of Theorem 1 in [3].Note that our model assumes the noise to be severely ill-posed; that is, the densityφ of noise ϵ satisfies |φ^(t)|~(1+|t|2)-β/2e-c|t|α(a.e.). Then it is reasonable to choose the Meyer scaling function as ϕ because the compact supportness of ϕ^ makes Kjϕ well defined, where
(58)(Kjϕ)(y):=12π∫ℝ‍eityϕ^(t)φ^(-2jt)dt.Compare with the proof of Theorem 1 in [3], the arguments of Theorem 9 are more complicated in the sense that we use Varshmov-Gilbert Lemma (Lemma 7). It is reasonable because we deal with unmatched estimation supf∈Br,qs(ℝ)E∥fn-f∥pp (p and r may not be equal), while they do the matched case supf∈B∞,∞s(ℝ)E∥fn-f∥∞.Although the Shannon functionϕS(t)=sinπt/πt is much simpler than the Meyer's, it cannot be used in our discussion because the Shannon’s does not belong to L(ℝ), while our theorems cover the case for p=1.Finally, it should be pointed out that we assume the independence of observationsY1,Y2,…,Yn in this paper. However, some dependent data are more important (of course, more complicated) in practice. We will investigate that case in future.



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