Guo and Liu Journal of Inequalities and Applications 2014, 2014:236 http://www.journalofinequalitiesandapplications.com/content/2014/1/236

RESEARCH

Open Access

Wavelet estimations for densities and their derivatives with Fourier oscillating noises Huijun Guo and Youming Liu* * Correspondence: [email protected] Department of Applied Mathematics, Beijing University of Technology, Pingle Yuan 100, Beijing, 100124, P.R. China

Abstract 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 W2s (L) and over L2 risk (Delaigle and Meister in Stat. Sin. 21:1065-1092, 2011). The current paper extends their theorem to Besov ball Bsr,q (L) and Lp risk with p, q, r ∈ [1, ∞] by using wavelet methods. We firstly show a linear wavelet estimation for densities in Bsr,q (L) over Lp risk, motivated by the work of Delaigle and Meister. Our result reduces to their theorem, when p = q = r = 2. 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 r ≤ p. In addition, our conclusions contain estimations for density derivatives as well. Keywords: Besov space; density derivative; density function; Fourier-oscillating noises; wavelet estimation

1 Introduction and preliminary One of the fundamental deconvolution problems is to estimate a density function fX of a random variable X, when the available data W , W , . . . , Wn are independent and identically distributed (i.i.d.) with Wj = Xj + δj

(j = , , . . . , n).

We assume that all Xj and δj are independent and the density function fδ of the noise δ is known.  Let the Fourier transform f ft of f ∈ L(R) be defined by f ft (t) = R f (x)eitx dx in this paper. ft When fδ satisfies  ft    f (t) ≥ c  + |t| –α δ

()

with c >  and α > , there exist lots of optimal estimations for fX [–]. However, many noise densities fδ have zeros in the Fourier transform domain, i.e., the inequality () does not hold. For example, Sun et al. described an experiment where data on the velocity of halo stars in the Milky Way are collected, and where the measurement errors are assumed to be uniformly distributed []. The classical kernel method provides a slower convergence rate in that case [–]. Delaigle and Meister [] developed a new method for a density fδ ©2014 Guo and Liu; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Guo and Liu Journal of Inequalities and Applications 2014, 2014:236 http://www.journalofinequalitiesandapplications.com/content/2014/1/236

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with   v  ft     f (t) ≥ csin πt   + |t| –α . δ   λ

()

Here c, λ, α >  and v ∈ N (non-negative integer set). Such noises are called Fourierft oscillating. Clearly, () allows fδ having zeros for v ≥ . When v = , () reduces to () (non-zero case). Delaigle and Meister defined a kernel estimator fˆn for a density fX in a Sobolev space and prove that with EX denoting the expectation of a random variable X,  sup E fX ∈Ws (L)

   s ˆfn (x) – fX (x) dx = O n– s+α+

b

()

a

under the assumption () (Theorem . in []). Here, Ws (L) stands for the Sobolev ball with the radius L. This above convergence rate attains the same one as in the non-zero case [–, ]. In particular, it does not depend on the parameter v. It seems that many papers deal with L estimations. However, Lp estimations ( ≤ p ≤ +∞) are important [, ]. On the other hand, Besov spaces contain many classical spaces (e.g., L Sobolev spaces and Hölder spaces) as special examples. The current paper extends () from Ws (L) to the Besov ball Bsr,q (L), and from L to Lp risk estimations. In addition, our results contain estimations for dth derivatives fX(d) of fX . The next section provides a linear wavelet estimation for fX(d) over a Besov ball Bsr,q (L) and over Lp risk (p, q, r ≥ ). It turns out that our estimation reduces to (), when d = , p = r = q = . Moreover, we show a nonlinear wavelet estimation which improves the linear one for r ≤ p in the last part.

1.1 Wavelet basis The fundamental method to construct a wavelet basis comes from the concept of multiresolution analysis (MRA []). It is defined as a sequence of closed subspaces {Vj } of the square integrable function space L (R) satisfying the following properties: (i) Vj ⊂ Vj+ , j ∈ Z (the integer set); (ii) f (x) ∈ Vj if and only if f (x) ∈ Vj+ for each j ∈ Z; (iii) j∈Z Vj = L (R) (the space j∈Z Vj is dense in L (R)); (iv) There exists ϕ(x) ∈ L (R) (scaling function) such that {ϕ(x – k)}k∈Z forms an orthonormal basis of V = span{ϕ(x – k)}k∈Z . j With the standard notation hj,k (x) :=   h(j x – k) in wavelet analysis, we can find a corresponding wavelet function ψ(x) =

(–)k+ h–k ϕ,k (x) with hk = ϕ, ϕ,k  k∈Z

such that, for a fixed j ∈ Z, {ψj,k }k∈Z constitutes an orthonormal basis of the orthogonal complement Wj of Vj in Vj+ []. Then each f ∈ L (R) has an expansion in L (R) sense, f=

k∈Z

αj,k ϕj,k +

∞



βl,k ψl,k ,

l=j k∈Z

where αj,k = f , ϕj,k , βl,k = f , ψl,k .

Guo and Liu Journal of Inequalities and Applications 2014, 2014:236 http://www.journalofinequalitiesandapplications.com/content/2014/1/236

A family of important examples are Daubechies wavelets DN (x), which are compactly supported in time domain []. They can be smooth enough with increasing supports as N gets large, although DN do not have analytic formulas except for N = . As usual, let Pj and Qj be the orthogonal projections from L (R) to Vj and Wj , respectively, Pj f =



αj,k ϕj,k ,

Qj f =

k∈Z



βj,k ψj,k = (Pj+ – Pj )f .

k∈Z

The following simple lemma is fundamental in our discussions. We use f p to denote Lp (R) norm for f ∈ Lp (R), and λ p do lp (Z) norm for λ ∈ lp (Z), where  {λ = {λk }, k∈Z |λk |p < +∞}, lp (Z) := {λ = {λk }, supk∈Z |λk | < +∞},

 ≤ p < ∞; p = ∞.

By using Proposition . in [], we have the following conclusion. Lemma . Let h be a Daubechies scaling function or the corresponding wavelet. Then there exists c ≥ c >  such that, for λ = {λk } ∈ lp (Z) and  ≤ p ≤ ∞,



   

≤ c j(  – p ) λ p . c j(  – p ) λ p ≤ λ h k j,k

k∈Z

p

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 Wpn (R) := {f ∈ Lp (R), f (n) ∈ Lp (R)} and f Wpn := f p + f (n) p . Then Lp (R) can be considered as Wp (R). For  ≤ p, q ≤ ∞, s = n + α with n ∈ N and α ∈ (, ], the Besov spaces are defined by     Bsp,q (R) := f ∈ Wpn (R), jα ωp (f (n) , –j ) j∈Z ∈ lq (Z) , with the associated norm f Bsp,q := f Wpn + {jα ωp (f (n) , –j )}j∈Z lq , where

ωp (f , t) := sup f (· + h) – f (· + h) + f (·) p |h|≤t

stands for the smoothness modulus of f . It should be pointed out that Bs, (R) = Ws (R). According to Theorem . in [], the following result holds. Lemma . Let ϕ = DN be a Daubechies scaling function with large N and ψ be the corresponding wavelet. If f ∈ Lp (R),  ≤ p, q ≤ ∞, s > , α,k = f , ϕ,k , and βj,k = f , ψj,k , then the following assertions are equivalent: (i) f ∈ Bsp,q (R); 



(ii) α,· lp + {j(s+  – p ) βj,· lp }j≥ lq < +∞; (iii) {js Pj f – f p }j≥ ∈ lq , where Pj is the projection operator to Vj .

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In each case,



     f Bsp,q ∼ α,· lp + j(s+  – p ) βj,· lp j≥ lq ∼ P f p + js Pj f – f p j≥ lq . Here and throughout, A  B denotes A ≤ CB for some constant C > ; A  B means B  A; we use A ∼ B standing for both A  B and B  A. Note that lp is continuously embedded into lp for p ≤ p . Then the above lemma implies that Bsp ,q (R) ⊂ Bsp ,q (R) for p ≤ p and s –

  = s – . p p

2 Linear wavelet estimation We shall provide a linear wavelet estimation for a compactly supported density function fX and its derivatives fX(d) under Fourier-oscillating noises in this section, motivated by the work of Delaigle and Meister. It turns out that our result generalizes their theorem. As in [], we define

p(x) =

v  

v m=

  πm (–)v–m fX x – . m λ

Then p ∈ L(R) and pft (t) = (ei

π t λ

()

ft

– )v fX (t). Delaigle and Meister found that

  πm , ηm p x – fX (x) = λ m= J

()

where J and ηm depend only on v and the support length of fX . Let ϕ = DN be the Daubechies scaling function with N large enough. Since both fX and ϕ have compact supports, the set Kj := {k ∈ Z : fX , ϕj,k  = } is finite and the cardinality |Kj | ∼ j . Then with αj,k = fX(d) , ϕj,k , Pj fX(d) =



αj,k ϕj,k .

k∈Kj

It is easy to see αj,k = (–)d fX , (ϕj,k )(d) . This with () and the Plancherel formula leads to  J    (–)d

i πλmt ft (d) ft ηm e p (t), (ϕj,k ) (t) . αj,k = π m= Note that pft (t) = (e

π it λ

ft

– )v

identity () reduces to (–)d αj,k = π



j

–  +dj ξ (t)

fW (t) ft

fδ (t)

()

j

–j

and [(ϕj,k )(d) ]ft (t) = –  +dj eik t [ϕ (d) ]ft (–j t). Then the

[ϕ (d) ]ft (–j t) ft fδ (t)

–j

ft

e–ik t fW (t) dt

()

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π mt π t ft where ξ (t) = Jm= ηm ei λ (ei λ – )v . Since the empirical estimator for fW is n nl= eiWl t , it is natural to define a linear wavelet estimator lin fˆn,d (x) =



αˆ j,k ϕj,k (x),

()

k∈Kj

with  (–)d αˆ j,k = n π n



j

–  +dj ξ (t)

[ϕ (d) ]ft (–j t) ft fδ (t)

l=

–j

e–ik t eiWl t dt.

()

lin When v = d = , p(x) = fX (x) and ηm = δm, . Then our estimator fˆn,d reduces to the clasft sical linear estimator for the case fδ having no zeros (see e.g., [–]). Let ψ be the Daubechies wavelet function corresponding to the scaling function DN and βj,k = f (d) , ψj,k . Similar to (), we define

 (–)d βˆj,k = n π n



j

–  +dj ξ (t)

[ψ (d) ]ft (–j t) ft fδ (t)

l=

–j

e–ik t eiWl t dt.

()

lin Then it can easily be seen that Eαˆ j,k = αj,k , Eβˆj,k = βj,k and Efˆn,d = Pj fX(d) . For a < b, L > , we consider the subset of Bsp,q (R),

   Bsp,q (a, b, L) := f ∈ Bsp,q (R) : f Bsp,q ≤ L, f (x) ≥ , f (x) dx = , supp f ⊂ [a, b] R

in this paper. ft

Lemma . Let ϕ = DN (N large enough), ψ be the corresponding wavelet and fδ satisfy  j (). If fX ∈ Bs+d r,q (a, b, L), r, q ∈ [, +∞], s > r and  ≤ n, then, for p ∈ [, +∞), p

E|αˆ j,k – αj,k |p  n–  jp(α+d) ,

p

E|βˆj,k – βj,k |p  n–  jp(α+d) .

Proof One shows only the first inequality; the second one is similar. Define Zl,k := Then αˆ j,k =

(–)d π  n



j

–  +dj ξ (t)

n

αˆ j,k – αj,k =

l= Zl,k

[ϕ (d) ]ft (–j t) ft fδ (t)

–j

e–ik t eiWl t dt.

and





(Zl,k – EZl,k ) := Yl,k . n n n

n

l=

l=

()

Clearly, EYl,k = . One estimates |Yl,k | and E|Yl,k | in order to use the Rosenthal in j  j (d) ft –j π t equality: By the assumption (), |Zl,k |  –  +dj |ei λ – |v |[ϕ ]ft ( t)| dt  –  +dj ( + |fδ (t)|  j +dj  j(α+d+  ) α (d) ft –j j α (d) ft  |t|) |[ϕ ] ( t)| dt =  ( + | t|) |[ϕ ] (t)| dt   ( + |t|)α |[ϕ (d) ]ft (t)| dt. Because ϕ = DN , the last integration is finite for large N . Hence, 

|Yl,k | ≤ |Zl,k |  j(α+d+  ) .

()

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s+d– 

 r Since fX ∈ Bs+d r,q (R) ⊂ B∞,q (R) (s > r ), fX ∞ < +∞ and fW ∞ = fX ∗ fδ ∞ ≤ fX ∞ × fδ  < +∞. This with the Parseval identity shows

 EZl,k

     (–)d j [ϕ (d) ]ft (–j t) –ik–j t ity  –  +dj  =  ξ (t) e dt  fW (y) dy e  ft π fδ (t)     – j +dj [ϕ (d) ]ft (–j t) –ik–j t    ξ (t) e  fW ∞   dt. ft fδ (t)

  Furthermore, one obtains EZl,k to (). Hence,



j

|–  +dj ( + |t|)α [ϕ (d) ]ft (–j t)| dt  j(α+d) thanks

  EYl,k = E|Zl,k – EZl,k | ≤ EZl,k  j(α+d) .

()

According to () and the Rosenthal inequality,  p p ∈ [, ), n–p ( n E|Yl,k | )  , E|αˆ j,k – αj,k |  –p nl= n p p –p   n E|Y | + n ( E|Y | ) , p ∈ [, +∞). l,k l,k l= l= p

p

Combining this with (), one obtains E|αˆ j,k – αj,k |p  n–  jp(α+d) for  ≤ p < , which is the desired conclusion. When p ≥ , p



 j(α+d+  )(p–) j(α+d) E|Yl,k |p  Yl,k p–  = j(  –+αp+dp) ∞ E|Yl,k |   p

p

due to () and (). Moreover, E|αˆ j,k – αj,k |p  n–p j(  –+αp+dp) + n–  jp(α+d) . Since j ≤ n, p

n–p j(  –+αp+dp)

p

–

=n

 j  p – p  jp(α+d) ≤ n–  jp(α+d) . n p

Finally, E|αˆ j,k – αj,k |p  n–  jp(α+d) . This completes the proof of the first part of Lemma ..  Now, we are in a position to state our first theorem. lin Theorem . Let fˆn,d be defined by ()-(), r, q ∈ [, +∞], p ∈ [, +∞) and s > r . Then with s := s – ( r – p )+ and x+ = max{, x},

sup fX ∈Bs+d r,q (a,b,L)

lin

p s p E ˆfn,d – fX(d) p  n– s +α+d+ .

lin lin – fX(d) p  fˆn,d – fX(d) r for r ≥ p, because Lr ([a, b]) is conProof It is easy to see that fˆn,d p p lin lin lin tinuously embedded into Lp ([a, b]). Moreover, E fˆn,d – fX(d) p  E fˆn,d – fX(d) r ≤ (E fˆn,d – p (d) r fX r ) r thanks to Jensen’s inequality. s +d When r ≤ p, s – p = s – r and Bs+d r,q (a, b, L) ⊂ Bp,q (a, b, L). Then

sup fX ∈Bs+d r,q (a,b,L)

lin

p E ˆfn,d – fX(d) p ≤

sup

s +d (a,b,L) fX ∈Bp,q

lin

p E ˆfn,d – fX(d) p .

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Therefore, it suffices to prove the theorem, for r = p, sup fX ∈Bs+d p,q (a,b,L)

lin

p sp E ˆfn,d – fX(d) p  n– s+α+d+ .

()

(d) s If fX ∈ Bs+d p,q (R), then fX ∈ Bp,q (R) and

(d) (d) p

Pj f – f  –jsp X X p

()

lin – Pj fX(d) = due to Lemma .. On the other hand, fˆn,d



ˆ j,k k∈Kj (α

– αj,k )ϕj,k and



p

lin p p – Pj fX(d) p  j(  –) E|αˆ j,k – αj,k |p  j  sup E|αˆ j,k – αj,k |p E ˆfn,d k∈Kj

k∈Kj

lin because of Lemma . and |Kj | ∼ j . This with Lemma . leads to E fˆn,d – Pj fX(d) p  p j ( n )  jp(α+d) . Combining this with (), one obtains p

lin

lin

p

p

p E ˆfn,d – fX(d) p  E ˆfn,d – Pj fX(d) p + Pj fX(d) – fX(d) p 

 j  p  jp(α+d) + –jsp . n



Take j ∼ n s+α+d+ . Then the inequality () follows, and the proof of Theorem . is finished.  s Remark . If p = q = r =  and d = , then s = s, Bs+d r,q (a, b, L) = W (a, b, L), Theorem . reduces to Theorem . in []. 

Remark . From the choice j ∼ n s+α+d+ in the proof of Theorem ., we find that our estimator is not adaptive, because it depends on the parameter s of Bsr,q (R). In order to avoid that shortcoming, we study a nonlinear estimation in the next part.

3 Nonlinear estimation This section is devoted to an adaptive nonlinear estimation, which also improves the convergence rate of the linear one in some cases. The idea of proof comes from []. Choose r > s, 

j ∼ n r +α+d+

and



j ∼ n α+d+ .

()

Let αˆ j,k and βˆj,k be defined by () and (), respectively, and 

βˆj,k , β˜j,k = ,

 |βˆj,k | > Tj(α+d) j/n, otherwise,

where the constant T will be determined in the proof of Theorem .. Then we define a nonlinear wavelet estimator non fˆn,d (x) :=

k∈Kj

αˆ j ,k ϕj ,k (x) +

j



j=j k∈Ij

β˜j,k ψj,k (x),

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where Kj := {k ∈ Z : fX , ϕj ,k  =  }, and Ij := {k ∈ Z : fX , ψj,k  =  }. Clearly, the cardinality |Ij | ∼ j since both fX and ψ have compact supports. Lemma . If jj ≤ n, then there exists c >  such that, for each T ≥ T > , 

P |βˆj,k – βj,k | >

 T j(α+d)   j/n  –c Tj . 

Proof By the definitions of βj,k and βˆj,k , βˆj,k – βj,k = (–)d Zl,k := π

 

j

–  +dj

J

ηm eit

π m λ



e

m=

Then EYl,k =  and with λ = T j(α+d) 

P |βˆj,k – βj,k | >

T j(α+d)  



π it λ

 n

n

 l= (Zl,k – EZl,k ) := n

n

l= Yl,k , where

v [ψ (d) ]ft (–j t) –i–j kt iW t – e l dt. e ft fδ (t)

j , n

    nλ j ≤  exp –  n (EYl,k + Yl,k ∞ λ/)

()

 + Yl,k ∞ λ/  thanks to the classical Bernstein inequality in []. On the other hand, EYl,k  j (α+d+  )j T j(α+d) j(α+d) j(α+d)  +  ≤ CT because of (), (), and jj ≤ n. Hence,  n nλ  + Y λ/) (EYl,k l,k ∞

≥

 j

n T n j(α+d) CTj(α+d)

=

T j, C

and () reduces to

   T T P |βˆj,k – βj,k | > j(α+d) j/n  e– C j = –c Tj  with c =

 C

log e. This completes the proof of Lemma ..



Theorem . Under the assumptions of Theorem ., there exist θ >  and T >  such that, for T ≥ T , sup fX ∈Bs+d r,q (a,b,L)



non

p E ˆfn,d – fX(d) p

⎧ sp ⎨(ln n)θ n– s+α+d+ ,

s p – (s–/r)+α+d+

⎩(ln n)θ n

(α+d+)p r ∈ ( s+α+d+ , p];

,

(α+d+)p r ∈ [, s+α+d+ ].

()

Proof Similar to [], one defines  μ := min

 s s , , s + α + d +  (s – /r) + α + d + 

and    ω := –sr + α + d + (p – r). 

()

Guo and Liu Journal of Inequalities and Applications 2014, 2014:236 http://www.journalofinequalitiesandapplications.com/content/2014/1/236

Page 9 of 15

(α+d+)p s It is easy to check that ω <  holds if and only if r > s+α+d+ , and μ = s+α+d+ as well (α+d+)p s as ω ≥  if and only if r ≤ s+α+d+ , and μ = (s–/r)+α+d+ . Then the conclusion of Theorem . can be rewritten as

sup fX ∈Bs+d r,q (a,b,L)

non

p E ˆfn,d – fX(d) p  (ln n)θ n–μp .

()

Choose j (s, r, q) and j (s, r, q) such that –μ

j (s,r,q)  n α+d+

μ

and

j (s,r,q)  n s .

Then it can easily be shown by () that j  j (s,r,q)  j (s,r,q)  j . Clearly,

 non

p

non  p  non  p (d) p

ˆf  Pj fˆn,d – fX(d) p + Dj ,j fˆn,d – fX(d) p + Pj fX(d) – fX(d) p , n,d – fX p

()

where

Dj ,j f =

j



βj,k ψj,k .

j=j k∈Z

By the assumption r ≤ p, s := s – ( r – p ) = s – s +d

s +d

 + p and Bs+d r,q (a, b, L) is continuously r

p (d) s fX ∈ Bp,q (R) and Pj fX(d) – fX(d) p  –j s p

embedded into Bp,q (a, b, L). Since fX ∈ Bp,q (R), thanks to Lemma .. This with j (s,r,q)  j and the definition of μ leads to

Pj f (d) – f (d) p  –j (s,r,p)s p  n–μp .  X X p

()

p p non non Note that Pj (fˆn,d – fX(d) ) = k∈Kj (αˆ j ,k – αj ,k )ϕj ,k . Then E Pj (fˆn,d – fX(d) ) p  j (  –) ×  p p ˆ j ,k – αj ,k |p  j  n–  j p(α+d) due to Lemma ., |Kj | ∼ j and Lemma .. By k∈Kj E|α 

–μ

j  j (s,r,q) and the choice j (s,r,q)  n α+d+ ,

 non  p – fX(d) p  E Pj fˆn,d



j (s,r,q) n

p/ j (s,r,q)p(α+d)  n–μp .

()

p non According to ()-(), it is sufficient to prove E Dj ,j (fˆn,d – fX(d) ) p  (ln n)θ n–μp : Define

   Bˆ j := k : |βˆj,k | > Tj(α+d) j/n ,    T Bj := k : |βj,k | > j(α+d) j/n ,     B j := k : |βj,k | > Tj(α+d) j/n ,

Sˆ j = Bˆ cj ; Sj = Bcj ; Sj = B cj .

j non ˆ ˆ ˆ – fX(d) ) = j=j Then Dj ,j (fˆn,d k∈Ij (βj,k – βj,k )[I{k ∈ Bj ∩ Sj } + I{k ∈ Bj ∩ Bj }]ψj,k –  j

ˆ ˆ j=j k∈Ij βj,k [I{k ∈ Sj ∩ Sj } + I{k ∈ Sj ∩ Bj }]ψj,k = ebs + esb + ebb + ess , where

ebs :=

j



j=j k∈Ij

(βˆj,k – βj,k )I{k ∈ Bˆ j ∩ Sj }ψj,k ,

esb :=

j



j=j k∈Ij

  βj,k I k ∈ Sˆ j ∩ B j ψj,k ,

Guo and Liu Journal of Inequalities and Applications 2014, 2014:236 http://www.journalofinequalitiesandapplications.com/content/2014/1/236

ebb :=

j



(βˆj,k – βj,k )I{k ∈ Bˆ j ∩ Bj }ψj,k ,

Page 10 of 15

ess :=

j=j k∈Ij

j



  βj,k I k ∈ Sˆ j ∩ Sj ψj,k .

j=j k∈Ij

In order to conclude Theorem ., one needs only to show that E ebs pp + E esb pp + E ebb pp + E ess pp  (ln n)θ n–μp .

()

j j j p p p By (), j – j ∼ ln n and j=j g  (j – j + )p– j=j gj p  (ln n)p– j=j gj p . This  j p   j with Lemma . shows that, for fˆ (x) = j=j k∈Ij fˆj,k ψj,k (x), there exists θ >  such that p

E fˆ pp  (ln n)θ sup j(  –) j ≤j≤j



E|fˆj,k |p .

()

k∈Ij

p To estimate E ebs p , one takes fˆj,k := (βˆj,k – βj,k )I{k ∈ Bˆ j ∩ Sj }. Then, for each k ∈ Bˆ j ∩ Sj ,   ˆ j := {k : |βˆj,k – βj,k | > T j(α+d) j }. |βˆj,k – βj,k | ≥ |βˆj,k | – |βj,k | > T j(α+d) j and Bˆ j ∩ Sj ⊂ D 



n

n

This with the Hölder inequality shows     ˆ j } ≤ E|βˆj,k – βj,k |p  EI{k ∈ D ˆ j}  . E|fˆj,k |p ≤ E|βˆj,k – βj,k |p I{k ∈ D  ˆ j } = P{|βˆj,k – βj,k | > T j(α+d) j }. Furthermore, using |Ij |  j , Lemma ., Clearly, EI{k ∈ D  n c T p p j –  jp(α+d) –  j ˆ and Lemma ., one obtains k∈Ij E|fj,k |   n   . Moreover, by (), p

p

E ebs pp  (ln n)θ sup n–  j(  +αp+dp–

c T  )

j ≤j≤j

p+αp+dp . Then p + αp + c c T p p p p –  j (  +αp+dp–  ) –  j  j p(α+d)

Choose T ≥ n

n







dp –

.

c T 

p

p

≤ , and supj ≤j≤j n–  j(  +αp+dp–

c T  )



. Similar to (), one has

E ebs pp  (ln n)θ n–μp .

()

In the proof of (), one needs to choose T ≥ c–  (p + αp + dp).  p

Now, one considers E esb p : For k ∈ Sˆ j ∩ Bj , |βˆj,k – βj,k | ≥ |βj,k | – |βˆj,k | > Tj(α+d) nj and  ˆ j := {k : |βˆj,k – βj,k | > T j(α+d) j }. By Lemma ., EI{k ∈ Sˆ j ∩ B j } ≤ EI{k ∈ Sˆ j ∩ B j ⊂ D  n

ˆ j }  –c Tj . Since fX(d) ∈ Bsr,q (R) ⊂ Bsp,q D (R), βj,· p := fX(d) , ψj,·  p  fX(d) s –j(s +/–/p) Bp,q

and



p   p

|βj,k |p EI k ∈ Sˆj ∩ B j  βj,· pp –c Tj  fX(d) s –j(s p+  –+c T) . Bp,q

k∈Ij

Moreover, it follows from the definition of esb and () that



E esb pp  (ln n)θ sup –j(s p+c T) ≤ (ln n)θ –j (s p+c T) . j ≤j≤j

By (), one can choose T ≥

j –j sp c j

(independent of n) so that j s p ≤ j (s p + c T). Hence, p

this above inequality reduces to E esb p  (ln n)θ –j s p . Similar arguments to () lead

Guo and Liu Journal of Inequalities and Applications 2014, 2014:236 http://www.journalofinequalitiesandapplications.com/content/2014/1/236

Page 11 of 15

to E esb pp  (ln n)θ n–μp .

()

In this above proof, one needs to choose T ≥ (c j )– (j – j )s p. p For E ebb p  (ln n)θ n–μp , one uses |Ij |  j , (), and Lemma . to find

j (s,r,q)

p



p 

I := E (βˆj,k – βj,k )I{k ∈ Bˆ j ∩ Bj }ψj,k  (ln n)θ sup n–  jp(α+d+  ) .

j ≤j≤j (s,r,q) j=j

k∈Ij

p

p

Recall that j ≤ j (s, r, q). Then I  (ln n)θ (n– j (s,r,q) )  j (s,r,q)p(α+d) , which reduces to I  (ln n)θ n–μp by similar arguments of (). It remains to show

j

p 





II := E (βˆj,k – βj,k )I{k ∈ Bˆj ∩ Bj }ψj,k  (ln n)θ n–μp .

j=j (s,r,q) k∈Ij

()

p

By Lemma . and the definition of Bj , k∈Ij E|(βˆj,k – βj,k )I{k ∈ Bˆj ∩ Bj }|p  k∈Ij E|βˆj,k –  p βj,k |p I{k ∈ Bj }  n–  jp(α+d) k∈Bj |βj,k T – –j(α+d) nj– |r . According to Lemma ., βj,· rr  fX(d) rBs –j(sr+r/–) . Hence, r,q

 p p–r r E(βˆj,k – βj,k )I{k ∈ Bˆj ∩ Bj }  n–  (ln n)θ –j(sr+  ––αp–dp+αr+dr) . k∈Ij

Combining this above inequality with (), one obtains II  (ln n)θ n–

p–r 

jω := An ,

sup j (s,r,q)≤j≤j

where ω := –sr + (α + d +  )(p – r) = –sr – r + p–r – 

When ω ≤ , An  (ln n) n θ

An  (ln n)θ n–

p–r 

–μ



j (s,r,q)ω

p 

+ αp + dp – αr – dr as defined in (). –μ

. By the choice j (s,r,q) ∼ n α+d+ , –μ

n α+d+ [–sr+(α+d+/)(p–r)] = (ln n)θ n–μp nr(μ– α+d+ s) .

–μ s for ω ≤ , μ – α+d+ s =  and An  (ln n)θ n–μp . Then one obtains the Since μ = s+α+d+ desired inequality (). (α+d+)p s When ω > , r < s+α+d+ and μ = (s–/r)+α+d+ . Take

p =

αp + dp + p –  . (s – r ) + α + d + 

 Then p–p = μp and r < p in that case. With fˆj,k = (βˆj,k – βj,k )I{k ∈ Bˆj ∩ Bj }, one knows from  Lemma . and the definition of Bj that

k∈Ij

p

E|fˆj,k |p ≤ n–  jp(α+d)

 p

 βj,k  p–p –j(α+d) n    (ln n)θ n–  j(p–p )(α+d) βj,· pp .  T  j

k∈Bj

Guo and Liu Journal of Inequalities and Applications 2014, 2014:236 http://www.journalofinequalitiesandapplications.com/content/2014/1/236

Page 12 of 15

Since r < p , βj,· p ≤ βj,· r . This with () leads to E ebb pp  (ln n)θ n–

j

 j( p–+αp+dp ) –j(α+d) p p sup   βj,· r  .

p–p 

j ≤j≤j

p–+αp+dp





p Note that  –j(α+d) = j(s– r +  ) due to the definition of p , as well as fX(d) ∈   Bsr,q (a, b, L) implies j(s– r +  ) βj,· r  fX(d) Bsr,q . Then

E ebb pp  (ln n)θ n–

p–p 

= (ln n)θ n–μp .

()

Finally, one estimates E ess p : Define fˆj,k := βj,k I{k ∈ Sˆ j ∩ Sj }. Then p



|fˆj,k |p ≤





|βj,k |

p–r

|βj,k | ≤ T r

j(α+d)

k∈Sj

k∈Ij

 p–r

r j   –j(s+  – r )r fX(d) Bs r,q n

due to r ≤ p and the definition of Sj . Using () and ω := –sr + (α + d +  )(p – r) in (), one obtains E ess pp

j

p  p–r 



(ln n)θ j ω n–  , ω ≤ ,

ˆ fj,k ψj,k = = E p–r

(ln n)θ j ω n–  , ω > . j=j k∈I 

When ω ≤ , E



j

()

p

j

ω ≤  holds if and only if μ = p–r j (s,r,q)ω n–  = n–μp . Hence,

βj,k I{k ∈ Sˆ j ∩ Sj }ψj,k p  (ln n)θ n– p

k∈Ij

j=j (s,r,q)

s s+α+d+

p–r 

j (s,r,q)ω . Recall that

–μ

and j (s,r,q) ∼ n α+d+ . Then it can be checked that

j

p 





 

βj,k I k ∈ Sˆ j ∩ Sj ψj,k  (ln n)θ n–μp . E

j=j (s,r,q) k∈Ij

()

p

On the other hand, Lemma . tells that

j (s,r,q)

p j (s,r,q) 





p

E βj,k ψj,k  j(  –) E|βj,k |p .



j=j

k∈Sˆ j ∩Sj

Since k ∈ Sj , |βj,k | ≤ Tj(α+d)

k∈Sˆ j ∩Sj

j=j

p



j n

and



p

k∈Sˆ j ∩Sj

|βj,k |p 



p

k∈Ij

jp(α+d) (ln n)θ n–   (ln n)θ ×

n–  j jp(α+d) . Moreover,

j (s,r,q)

p



 

βj,k I k ∈ Sˆ j ∩ Sj ψj,k E

j=j

k∈Ij

p

p

 (ln n)θ n– 

p –

 (ln n)θ n

sup j ≤j≤j (s,r,q)

p

j(  –) j jp(α+d) 

j (s,r,q)(α+d+  )p  (ln n)θ n–μp ,

()

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Page 13 of 15

–μ

where the last inequality comes from the choice j (s,r,q) ∼ n α+d+ . Combining this with (), one has, for ω ≤ , E ess pp  (ln n)θ n–μp .

() p

It remains to show E ess p  n–μp for ω > . By Lemma .,





p /p  j(  – p )

  ˆ ˆ βj,k I{k ∈ Sj ∩ Sj }ψj,k ∼  βj,k I{k ∈ Sj ∩ Sj }

p

k∈Ij

k∈Ij

≤

j(  – p )

βj,· lp .

This with Lemma . shows that 

 j



q /q

βj,k I{k ∈ Sˆ j ∩ Sj }ψj,k 

js



j



≤

p

k∈Ij

j=j (s,r,q)



j(s +  – p )

βj,· lp

q

/q

≤ fX(d) Bs .

()

p,q

j=j (s,r,q)

When q = ,

j



ˆ β I{k ∈ S ∩ S }ψ j,k j j j,k

p

j=j (s,r,q) k∈Ij j





–j (s,r,q)s

 ˆ  βj,k I{k ∈ Sj ∩ Sj }ψj,k  –j (s,p,q)s .

 js

p

k∈Ij

j=j (s,r,q)

When q = +∞,

j



ˆ βj,k I{k ∈ Sj ∩ Sj }ψj,k

p

j=j (s,r,q) k∈Ij j





–js fX(d) Bs  (ln n)θ –j (s,p,q)s . p,q

j=j (s,r,q)

For  < q < +∞, by the Hölder inequality,

j



ˆ βj,k I{k ∈ Sj ∩ Sj }ψj,k

p

j=j (s,r,q) k∈Ij

 

j



j=j (s,r,q)

 –j (s,r,q)s .

–js q

   q

 j

j=j (s,r,q)



q  q

βj,k I{k ∈ Sˆ j ∩ Sj }ψj,k 

js

k∈Ij

p

Guo and Liu Journal of Inequalities and Applications 2014, 2014:236 http://www.journalofinequalitiesandapplications.com/content/2014/1/236

Page 14 of 15

μ

Using () and the choice j (s,r,q) ∼ n s , one obtains



p

j

θ –μp

ˆ E βj,k I{k ∈ Sj ∩ Sj }ψj,k

 (ln n) n . p

j=j (s,r,q) k∈Ij

On the other hand,

j

p

(s,r,q)

p–r θ j (s,r,q)ω – 

ˆ E βj,k I{k ∈ Sj ∩ Sj }ψj,k n

 (ln n)  j=j

p

k∈Ij

μ

thanks to (). According to the choice of j (s,r,q) ∼ n s and μ=

s – /r + /p s = (s – /r) + α + d +  (s – /r) + α + d +  p–r

p

for ω > , one finds j (s,r,q)ω n–   n–μp by direct computations. Hence, E ess p  n–μp in each case, which is (). Now, () follows from (), (), (), and (). The proof is done.  Remark . We find easily from Theorem . and Theorem . that the nonlinear wavelet estimator converges faster than the linear one for r ≤ p. Moreover, the nonlinear estimator is adaptive, while the linear one is not. Remark . 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.

Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript. Acknowledgements This paper is supported by the National Natural Science Foundation of China (No. 11271038). Received: 24 March 2014 Accepted: 28 May 2014 Published: 08 Jun 2014 References 1. Walter, GG: Density estimation in the presence of noise. Stat. Probab. Lett. 41, 237-246 (1999) 2. Pensky, M, Vidakovic, B: Adaptive wavelet estimator for nonparametric density deconvolution. Ann. Stat. 27, 2033-2053 (1999) 3. Fan, J: On the optimal rates of convergence for nonparametric deconvolution problems. Ann. Stat. 19, 1257-1272 (1991) 4. Fan, J, Koo, J: Wavelet deconvolution. IEEE Trans. Inf. Theory 48, 734-747 (2002) 5. Lounici, K, Nickl, R: Global uniform risk bounds for wavelet deconvolution estimators. Ann. Stat. 39, 201-231 (2011) 6. Li, R, Liu, Y: Wavelet optimal estimations for a density with some additive noises. Appl. Comput. Harmon. Anal. 36, 416-433 (2014) 7. Sun, J, Morrison, H, Harding, P, Woodroofe, M: Density and mixture estimation from data with measurement errors. Technical report (2002) 8. Devroye, L: Consistent deconvolution in density estimation. Can. J. Stat. 17, 235-239 (1989) 9. Hall, P, Meister, A: A ridge-parameter approach to deconvolution. Ann. Stat. 35, 1535-1558 (2007) 10. Meister, A: Deconvolution from Fourier-oscillating error densities under decay and smoothness restrictions. Inverse Probl. 24, 1-14 (2008)

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11. Delaigle, A, Meister, A: Nonparametric function estimation under Fourier-oscillating noise. Stat. Sin. 21, 1065-1092 (2011) 12. Donoho, DL, Johnstone, IM, Kerkyacharian, G, Picard, D: Density estimation by wavelet thresholding. Ann. Stat. 24, 508-539 (1996) 13. Hernández, E, Weiss, G: A First Course on Wavelets. CRC Press, Boca Raton (1996) 14. Daubechies, I: Ten Lectures on Wavelets. SIAM, Philadelphia (1992) 15. Härdle, W, Kerkyacharian, G, Picard, D, Tsybakov, A: Wavelets, Approximation, and Statistical Applications. Lecture Notes in Statist. Springer, New York (1998)

10.1186/1029-242X-2014-236 Cite this article as: Guo and Liu: Wavelet estimations for densities and their derivatives with Fourier oscillating noises. Journal of Inequalities and Applications 2014, 2014:236

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