A maxiset approach of a Gaussian noise model

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Preprint submitted on 7 Jun 2005

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Christophe Chesneau

To cite this version:

ccsd-00004331, version 9 - 7 Jun 2005

(nondenitive version)

Chesneau Christophe

June 7, 2005

Laboratoire de Probabilités et Modèles Aléatoires, Université Pierre et Marie Curie, 175 rue de Chevaleret,F-75013Paris,Fran e. hristophe. hesneau4wanadoo.fr

Abstra t

We onsider the problem of estimating an unknown fun tion

f

in a homos edasti Gaussian white noise setting under

L

p

risk. The parti ularity of this modelis that it has anintermediate fun tion, say

v

,whi h ompli atestheestimatesigni antly. Whilevaryingtheassumptionson

v

, weinvestigatetheminimaxrateof onvergen eovertwoballsofspa eswhi hbelongto familyof Besov lasses. Oneisdenedasusual andtheother alled'weightedBesov balls'used

v

expli itly. Adoptingthe maxisetapproa h,wedevelop anaturalhardthresholdingpro edure whi hattained theminimaxrateof onvergen ewithinalogarithmi fa torovertheseweightedballs.

Keywords: Minimax, Mu kenhouptweights, Maxiset,Gaussian noise, warped wavelets, wavelet thresholding.

AMS 1991Subje t Classi ation: Primary: 62G07,Se ondary: 62G20,42B20.

1 Introdu tion

ConsidertheGaussianwhitenoisemodelinwhi hweobservepro esses

Y

t

governedby

dY

t

= H

v

(f )(t)dt +

1

√

n

dW

t

,

n

∈ N

∗

_,

_t

_{∈ [0, 1]}

(1)

wheretheoperator

H

v

:

B([0, 1]) → L

2

_{([0, 1])}

isdened by

H

v

(f )(t) =

f

(t)

v(t)

,

with

B([0, 1]) = {f measurable on [0, 1], sup

x∈[0,1]

|f(x)| < ∞}

and

L

2

_{([0, 1]) =}

_{{f measurable on [0, 1], kfk}

2

₂

₌

Z

1

0

|f(x)|

2

_{dx <}

∞}.

Thefun tion

v

issupposedtobeknownandtosatisfythe ondition'

1

v

belongsto

L

2

_{([0, 1])}

'. Thepro ess

W

t

is astandard Brownian motionon

[0, 1]

. The fun tion

f

is theunknown fun tion of interest. We wanttore onstru t

f

fromtheobservations

{

R

1

0

h(t)dY

t

, h

∈ L

2

_{([0, 1])}

_}

.

In thesimplest ase where

v

is onstant, weobservethe well known Gaussian whitenoise model whi h has been onsidered in several papers starting from Ibragimov and Has'minskii (1977). Under ertainassumptions onthesmoothness of

f

,themodel(1) be omes anappropriatelargesamplelimit tomoregeneralnonparametri modelssu hasprobabilitydensityestimation(seeNussbaum(1996))or nonparametri regression(seeBrownandLow(1996)).Minimaxpropertiesinvariousriskovernumerous fun tion spa es an befoundinthebookofTsybakov(2004).

Inthe asewhere

v

isspatially inhomogeneous,the urveestimationis signi antlymore ompli- ated. Forinstan e, onsidertheobservationofdata

(Y

1

, X

1

), ..., (Y

n

, X

n

)

where

wheretherandomvariables

X

i

arei.i.d,independentof

ǫ

i

,withdensity

g

. The

ǫ

i

'sarenormali.i.dwith meanzeroand varian eone. BrownandLow(1996)haveshownthatif

σ

and

g

satisfysome ondition ofboundednessand

f

belongsto ertainSobolev lassesthenthemodel

dZ

t

= v(t)dY

t

,

t

∈ [0, 1]

(3) isasymptoti allyequivalent(inLeCam'ssense)to (2)under the alibration

v

=

σ

√

_g

. Anappli ationof this result anbefoundin Efromovi hand Pinsker(1996). Forother equivalen es on erning(3), see GramaandNussbaum(1998).

In this paper, we are fo used on the model (1) for general

v

and we take the problem in the frameworkofwavelet analysis. Wewish toestimate

f

on

[0, 1]

byameasurable fun tionon

[0, 1]

with respe ttotheobservations

{

R

1

0

h(t)dY

t

, h

∈ L

2

_{([0, 1])}

}

under the

L

p

risk

E

_f

₍

Z

1

0

| ˆ

f

(t)

_{− f(t)|}

p

_dt),

_p

≥ 1.

Wehavedenoted

E

f

theexpe tationwithrespe tthedistribution

P

f

ofpro esses

Y

t

. Ourstudy anbe dividedintwoparts.

Inarst part,weinvestigatetheestimation of

f

overusualBesovballs

B

s,π,r

(L)

. Weshowthat theminimaxpropertiesobtainforthe asewhere

v

isbounded fromaboveandbelow anbeextended (without deteriorating the rate of onvergen e) to more generalfun tions

v

. More pre isely, we show that if

π

≥ p ≥ 1

and

v

belongsto

L

π

′

_{([0, 1])}

for

π

′

_{= max(π, 2)}

thentheminimaxrateof onvergen e isoftheform

n

−α

1

p

_{where α}

1

=

s

1 + 2s

.

Forothervaluesontheparameters(

s, π, r

),weshowthatif

v

isboundedfromabovethentheminimax rateof onvergen eover

B

s,π,r

(L)

isoftheform

(

ln(n)

n

)

α

2

p

_{where α}

2

=

s

₋

_π

1

+

1

_p

2(s

−

1

π

) + 1

.

Inthe asewhere

π

≥ p > 2

,itisnaturaltoaddressthefollowingquestion: anweobtainthesame minimaxrateoversu hspa esforany

v

whi hdoesnotbelongto

L

π

_{([0, 1])}

? Usinganexpli itexample, weshowthattheansweris'No'.

This result motivates us to devote a se ond part in whi h we investigate other fun tion spa es more adapted to our model. Our hoi e will be made on Besov balls onstru ted on a wavelet basis warpedby afa tor depending on

v

. Su h spa es were introdu ed in analysis byQui (1982) andwere re ently developed in statisti sby Kerkya harianand Pi ard(2005). These authors have established good estimation results in a regression setting with random design (i.e (2) with

σ(.) = 1

) for very generaldensities

g

. Thekeyofthesu essofourstudyrestsonthefollowingargument: under ertain onditionsonthewarpingfa torwhi hrefertoMu kenhoupttheory,thewarpedwaveletbases possess someinterestinggeometri alpropertiesin the

L

p

norm whi h allowus to onsider fun tion spa esand pro eduresdeeplylinkedtothemodel.

Using these analyti al tools, we show that if

π

≥ p > 1

and if

v

is subje t to a property of Mu kenhoupttypethentheminimaxrateoverweightedBesovballs

B

G

s,π,r

(L)

denedstartingfrom

G

, theprimitiveof

1

v

2

, isoftheform

n

−α

1

_{where α}

1

=

s

1 + 2s

for

s

largeenough. Thehypothesesmadeon

v

aremoregeneralthan onditionsofboundednessorother onditionsof integrabilitydepending dire tlyoftheparameter

π

.

Finally, weuse this warped waveletbasis to onstru tanaturalpro edure whi h stayas lose as possibletothestandardhardthresholdingalgorithm. Inordertomeasurehisperforman eunder

L

p

risk, weisolatetheasso iatedmaxiset. Thisstatisti altooldevelopedbyCohen,DeVore,Kerkya harianand Pi ard(2000) onsists in investigating themaximal spa e (ormaxiset) where apro edure has agiven rateof onvergen e. Oneofthemainadvantagesofthisapproa histoprovideafun tionalsetwhi his authenti ally onne tedto thepro edureandthemodel. Thus,by hoosingtherate

(ln(n))

α

1

_n

−α

1

_{where α}

1

=

s

1 + 2s

and onsidering

π

≥ p > 1

,weprovethatourweightedBesovball

B

G

s,π,r

(L)

isin ludedin themaxiset of our pro edure. So, we on lude that it is 'near to the optimal' i.e it attains the minimax rate of onvergen euptoalogarithmi fa tor.

Thepaperisorganizedasfollows.

Se tion 2 denes the basi tools (Mu kenhoupt weights, warped wavelet basis ...), inequalities and fun tion spa es weshall need in the study. In Se tion 3 we investigate the minimax rate over usual Besovballs. Se tion4dothesamestudybutovertheweightedBesovballs. Se tion5isdevotedtothe performan eofanaturalhardthresholdingpro edurewhentheunknownfun tionofinterestbelongsto these weighted spa es. InSe tion 6,wedes ribeanotherstatisti al model andweexplainwhywe an haveresults similar to those obtained for (1). Finally, Se tion 7 is devoted to the proofs of te hni al lemmas.

2 Mu kenhoupt ondition, warped wavelet bases and fun tion spa es

Throughoutthispaper,foraweight

m

(i.enonnegativelo allyintegrablefun tion) on

[0, 1]

,weset

L

p

m

([0, 1]) =

f measurable on [0, 1] | kfk

p

m,p

=

Z

1

0

|f(t)|

p

_{m(t)dt <}

∞

where

L

p

_{([0, 1]) = L}

p

1

([0, 1])

denotestheusualLebesguespa ei.e

L

p

_{([0, 1]) =}

f measurable on [0, 1] | kfk

p

_p

₌

Z

1

0

|f(t)|

p

_{dt <}

∞

.

2.1 Mu kenhoupt weight

Firstre allthenotionofMu kenhouptweight. Denition 2.1. Let

1 < p <

∞

and

q

su h that

1

p

+

1

p

= 1.

A weight

m

satises the

A

p

- ondition (or belongs to

A

p

) i there exists a onstant

C >

0

su h that for any measurable fun tion

h

and any subinterval

I

of

[0, 1]

wehave

(

1

|I|

Z

I

|h(x)|dx) ≤ C(

1

m(I)

Z

I

|h(x)|

p

_m(x)dx)

1

p

(4)

where

|I|

denotesthe Lebesguemeasureof

I

and

m(I) =

R

I

m(x)dx

.

If

m

veries the

A

p

onditionthenitisaMu kenhoupt weight. Example2.1. The weight

m(x) = x

σ

satisesthe

A

p

- ondition with

p >

1

i

−1 < σ < p − 1

. Letusintrodu eoneofthemostinterestingpropertyrelatedtothisnotion.

Lemma 2.1. Let

1 < p <

∞

. If

w

satises the

A

p

onditionthen there existsa onstant

C >

0

su h that forany subintervals

S

⊆ B ⊆ [0, 1]

wehave

w(B)(

|S|

|B|

)

p

≤ Cw(S).

Proof of Lemma 2.1. Itsu estoapply(4)withthefun tion

h

= 1

S

andtheinterval

I

= B

.

The previous ondition hasbeen introdu ed by Mu kenhoupt (1972)and widely used afterwards inthe ontextofCalderón-Zygmundtheory. The

A

p

- ondition hara terizestheboundednessof ertain integraloperatorson

L

p

m

spa es liketheHardy-Littlewoodmaximal operatorortheHilberttransform. Forthe ompletetheory,see thebookofStein(1993).

2.2 Warped wavelet bases and Mu kenhoupt weights

Firstweintrodu ethewarpedwaveletbases.Se ondwesetsomeresultswhi h willbeintensivelyused inthesequelofthispaper.

Let

N

beanintegerstri tlypositive. Wedenoteby

ξ

T

=

{φ

τ,k

(T (.)), k

∈ ∆

τ

; ψ

j,k

(T (.)); j

≥ τ, k ∈ ∆

j

}, ∆

j

=

{0, ..., 2

j

− 1},

thewarped wavelet basisadaptedontheinterval

[0, 1]

onstru tedstartingfrom

• ψ

thewaveletasso iatedwithamultiresolutionanalysisontheline

V

j

=

{φ

j,k

, k

∈ Z}

su h that

Supp(φ) = Supp(ψ) = [

_{−N + 1, N]}

and

R ψ(t)t

l

_dt

_{= 0}

for

l

= 0, ..., N

− 1

. Letus re allthat on theunit intervalthereexists aninteger

τ

satisfying

2

τ

_{≥ 2N}

su hthatone anbuiltatea hlevel

j

_{≥ τ}

awaveletsystem

(φ

j,k

, ψ

j,k

)

where

φ

j,k

(x) = 2

j

2

φ(2

j

_x

− k), k = N, N, N + 1, ..., 2

j

− N − 1

and

ψ

j,k

(x) = 2

j

2

ψ(2

j

x

− k), k = N, N, N + 1, ..., 2

j

− N − 1.

Forea hfun tions,weadd

N

fun tionsontheneighborhoodof

0

whi hhavethesupport ontained in

[0, (2N

− 1)2

−j

_]

and

N

fun tions on theneighborhoodof

1

whi h havethesupport ontained in

[1

− (2N − 1)2

−j

_,

_1]

. Forsimpli ity,wedenoteby"

τ

− 1

"theintegersu hthat

ψ

τ

−1,k

= φ

τ,k

.

• T

ameasurablefun tion on

[0, 1]

whi hareanknown,in reasing,bije tive,absolutely ontinuous andsatises

T

(0) = 0 and T (1) = 1.

Weasso iatetothisfun tiontheweight

w(.) =

1

˜

T

(T

−1

_(.))

(5)

where

T

˜

denotes the derivative of

T

and

T

−1

its inverse fun tion. Remark that for any measurable positivefun tion

z

dened on

[0, 1]

,

w

satises

Z

1

0

z(T (x))dx =

Z

1

0

z(x)w(x)dx.

Notethatthewarpedwaveletbases anbeviewedasageneralizationoftheregularwaveletbases. See Meyer(1990)andDaube hies(1992)forwaveletbasesontherealline. SeeCohen,Daube hies,Jawerth andVial(1992)forwaveletbases ontheinterval.

Let

∞ > p > 1

. If

w

veriesthe

A

p

onditionthen, forany

ν

≥ τ

,anyfun tion

f

of

L

p

_{([0, 1])}

an bede omposedon

ξ

T

as

f

(x) = P

ν

T

(f )(x) +

X

j

≥ν

X

k∈∆

j

β

T

_j,k

ψ

j,k

(T (x)),

where

P

_j

T

(f )(x) =

X

k∈∆

j

α

T

_j,k

φ

j,k

(T (x)), α

T

j,k

=

Z

1

0

f

(T

−1

(t))φ

j,k

(t)dt

and

β

T

_j,k

=

Z

1

0

f

(T

−1

(t))ψ

j,k

(t)dt.

Letusre allsomepropertieslinkedto

ξ

T

.

Property 2.1. Let

v >

0

. Thereexistsa onstant

C >

0

su hthat

X

k

∈∆

j

|φ

j,k

(T (x))

|

v

≤ C2

jv

2

,

x

∈ [0, 1].

Property 2.2. If

w

∈ A

p

thenthere existtwo onstant

c >

0

and

C >

0

su hthat for

j

≥ τ

wehave

c2

jp

2

X

k

∈∆

j

|α

T

j,k

|

p

w(I

j,k

)

≤ kP

j

T

(f )

k

p

p

≤ C2

jp

2

X

k

∈∆

j

|α

T

j,k

|

p

w(I

j,k

)

wherewehave set

I

j,k

= [

k

2

j

,

k+1

2

j

]

. These inequalities arealways trueif weex hanged

φ

by

ψ

.

2.3 Fun tion spa es

Firstly, letuspre ise thatweightedBesovballs

B

T

s,π,r

(L)

hasthree parameters:

s

measures thedegree ofsmoothness,

π

and

r

spe ifythe typeofnorm used tomeasure thesmoothness. Letus dene these spa eswithpre isedetailsbelow.

For any measurable fun tion

f

dened on

[0, 1]

, denote the asso iated

N

-th order modulus of smoothness as

ρ

N

(t, f, T, π) = sup

|h|≤t

Z

J

N h

|

N

X

k=0

N

k



(

−1)

k

_f

_(T

−1

_{(T (u) + kh))}

_|

π

_du

!

1

π

where

J

N h

=

{x ∈ [0, 1] : T (x) + Nh ∈ [0, 1]}

. Let

N > s >

0

,

∞ ≥ π, r > 1

. Wesaythat afun tion

f

of

L

π

_{([0, 1])}

belongsto theweightedBesovballs

B

T

s,π,r

(L)

i

Z

1

0

 ρ

N

_{(t, f, T, π)}

t

s



r

₁

t

dt



1

r

≤ L < ∞

withtheusualmodi ationif

r

=

∞

. Thesespa es anbeviewedasageneralizationofusualBesovballs. If we are in the ase where

T

= I

d

then we simply denote

ξ

T

_{= ξ}

,

α

T

j,k

= α

j,k

,

β

T

j,k

= β

j,k

,

P

T

j

(f ) = P

j

(f )

and

B

T

s,π,r

(L) = B

s,π,r

(L)

.

Startingfrom thepreviousdenition,we ansetalistofpropertieswhi hlinktheweightedBesov ballswiththewarpedwaveletbasisontheunit interval. Seebelowthreeofthese.

Property 2.3. If

w

∈ A

π

wehave

f

_{∈ B}

_s,π,r

T

(L) =

_{⇒ (}

X

j≥τ −1

(2

j(s+

1

2

)

(

X

k∈∆

j

|β

T

j,k

|

π

w(I

j,k

))

1

π

)

r

)

1

r

≤ L.

(6) for

π

≥ 1

,

r

≥ 1

and

N > s >

0

.

Moreover,wehavethere ipro ityfor

s

largeenough. Property 2.4. If

w

∈ A

π

wehave

f

_{∈ B}

_s,π,r

T

(L)

_{⇐= (}

X

j

≥τ −1

(2

j(s+

1

2

)

(

X

k

∈∆

j

|β

T

j,k

|

π

w(I

j,k

))

1

π

)

r

)

1

r

≤ L.

(7)

for

π

≥ 1

,

r

≥ 1

and

N > s

≥ q(w)

where

q(w) =

(

inf

v>1

{w satisfies the A

v

condition

} if w is not a constant on [0, 1],

0

if w is constant on [0, 1].

(8)

Thefollowingpropertyissimilarto Property2.3butexpressedin termof

P

j

(f )

. Property 2.5. If

w

∈ A

π

wehave

f

_{∈ B}

T

s,π,r

(L) =

⇒ (

X

j

≥τ

(2

js

kP

T

j

(f )

− fk

π

)

r

)

1

r

≤ L

(9)

These results are always true with the usual modi ation if

r

=

∞

. Forfurther details on this subse tion,wereferthereaderto thearti leofKerkya harianandPi ard(2005).

In the sequel, the onstants

C

,

C

′

,

C

′′

,

c

,

c

′

,

c

′′

3 Minimax study over Besov balls

Letusre allthatweobservethemodel(1)underthetwofollowingassumptionson

v

and

f

:

•

1

v

belongsto

L

2

_{([0, 1])}

,

• kfk

∞

<

∞.

Letussetthefollowingnotations:

ǫ

= sπ

₋

p

− π

2

,

α

1

=

s

1 + 2s

and α

2

=

s

₋

1

_π

+

1

_p

2(s

₋

_π

1

) + 1

.

Weshallexhibittheminimaxrateof onvergen eoverusualBesovballsforseveralvaluesof

ǫ

. Therst partofthisse tion isdevotedtotheproofsofthetwofollowingtheorems.

Theorem3.1. Let

∞ > p ≥ 1

and

∞ ≥ π ≥ p

. Assumethat

v

satises thefollowing ondition:

v

_{∈ L}

π

′

([0, 1])

(10)

where

π

′

_{= max(π, 2)}

. Then for

N > s >

0

and

∞ ≥ r ≥ 1

wehave

inf

ˆ

f

sup

f

∈B

s,π,r

(L)

E

_f

₍

_{k ˆ}

f

_{− fk}

p

_p

)

≍ n

−α

1

p

_.

Theorem3.2. Let

∞ > p > 1

. Assumethat

v

satisesthe following ondition:

kvk

∞

<

∞.

(11) Then for

N > s >

1

π

,

∞ ≥ π ≥ 1

,

∞ ≥ r ≥ 1

and

ǫ <

0

wehave

inf

ˆ

f

sup

f∈B

s,π,r

(L)

E

_f

₍

_{k ˆ}

f

_{− fk}

p

_p

)

≍ (

ln(n)

_n

)

α

2

p

For

ǫ

= 0

,there exist

C >

0

and

c >

0

satisfying

c(

ln(n)

n

)

α

2

p

≤ inf

_ˆ

f

sup

f

∈B

s,π,r

(L)

E

_f

₍

_{k ˆ}

_f

_{− fk}

p

_p

₎

_{≤ C(}

ln(n)

n

)

α

2

p

_(ln(n))

(

p

2

−

π

r

)

+

_.

Remark3.1. Forthe twolower bounds, onlythe ondition

1

v

∈ L

2

_{([0, 1])}

isdeterminant.

3.1 Proof of Theorem 3.1: upper bound and lower bound 3.1.1 Upper bound

Here,weusethestandardmethodwhi h onsists in representingtheunknown fun tion

f

on aregular waveletbasisandinstudyingtheupperboundattainedbytheasso iatedlinearwaveletpro edure. Theorem3.3. Let

∞ > p ≥ 1

and

π

≥ p

. Assumethatthe ondition(10) holds. Consider

f

ˆ

l

thelinear estimator denedby

ˆ

f

l

(x) =

X

k∈∆

j0

ˆ

α

j

0

,k

φ

j

0

,k

(x)

(12) where

ˆ

α

j

0

,k

=

Z

1

0

φ

j

0

,k

(t)v(t)dY

t

.

Then for

N > s >

0

and

∞ ≥ r ≥ 1

thereexistsa onstant

C >

0

su hthat

sup

f∈B

s,π,r

(L)

E

_f

₍

_{k ˆ}

f

l

_{− fk}

p

_p

)

≤ Cn

−α

1

p

for

j

0

the integersatisfying

2

j

0

_{≃ n}

1+2s

1

Proof of Theorem 3.3. UsingHölder'sinequality, for

π

≥ p

wehave

E

_f

₍

_{k ˆ}

_f

l

_{− fk}

p

_p

₎

_{≤ C(E}

_f

₍

_{k ˆ}

_f

l

_{− fk}

π

_π

₎₎

p

π

.

(13) UsingMinkowski'sinequalityandtheelementaryinequality

(

|x + y|)

π

≤ 2

π

−1

₍

_|x|

π

₊

|y|

π

_{), x, y}

∈ R,

the

L

π

riskof

f

ˆ

anbede omposed asfollows:

E

_f

₍

_{k ˆ}

_f

l

_{− fk}

π

π

)

≤ C(E

f

(

k ˆ

f

l

− P

j

0

(f )

k

π

π

) +

kP

j

0

(f )

− fk

π

π

)

= C(S

1

+ S

2

).

(14)

Sin e

f

∈ B

s,π,r

(L)

⊂ B

s,π,

∞

(L)

,Property2.5givesus

S

2

≤ L2

−j

0

sπ

.

(15)

Usingthedenitionof

f

ˆ

l

andProperty2.2,onegets

S

1

=

E

f

(

k

X

k∈∆

j0

(ˆ

α

j

0

,k

− α

j

0

,k

)φ

j

0

,k

(.)

k

π

π

)

≤ C(2

j

0

(

π

2

−1)

X

k∈∆

j0

E

_f

₍

_|ˆ

α

j

0

,k

− α

j

0

,k

|

π

₎₎

=

C2

j

0

(

π

2

−1)

S

∗

1

.

(16)

Letus onsider

ρ

j,k

denedby

ρ

j,k

=

s

Z

1

0

v

2

_(t)φ

2

j,k

(t)dt.

(17) Wehave learly

ˆ

α

j

0

,k

− α

j

0

,k

=

1

√

_n

Z

1

0

v(t)φ

j

0

,k

(t)dW

t

∼ ρ

j

0

,k

ǫ

n

where

ǫ

n

israndomvariablesu hthat

ǫ

n

∼ N (0,

1

n

).

Tostudy

S

∗

1

, weonlyneedthese ond pointofthefollowinglemmawhi h willbeprovedinAppendix. (The rstpointwill beusedlaterinthestudy.)

Lemma 3.1. Let

n

∈ N

∗

. If

V

n

∼ N (0,

1

n

)

then for

κ

≥ 2

√

2π

there exists a onstant

C >

0

only dependingon

p

su hthat

• P

f

(

|V

n

| ≥

κ

₂

q

ln(n)

n

)

≤ Cn

−

π

2

,

• E

f

(

|V

n

|

π

)

≤ Cn

−

π

2

.

Thus,onegets

S

1

∗

≤ Cn

−

π

2

X

k

∈∆

j0

ρ

π

_j

₀

_,k

.

(18)

First onsider the asewhere

2 > π

≥ 1

. Hölder'sinequality,Property2.1and ondition(10)yield

X

k

∈∆

j0

ρ

π

_j

₀

_,k

_{≤ (}

X

k

∈∆

j0

Z

1

0

v

2

(t)φ

2

j

0

,k

(t)dt)

π

2

(

X

k

∈∆

j0,k

1)

1−

π

2

≤ C(2

j

0

Z

1

0

v

2

(t)dt)

π

2

2

j

0

(1−

π

2

)

≤ C

′

2

j

0

_.

(19)

Se ond investigate the ase where

∞ ≥ π ≥ 2

. Applying Hölder'sinequality with the measure

dν

=

φ

2

_j

₀

_,k

(t)dt

,usingProperty2.1and ondition(10),onegets

X

k∈∆

j0

ρ

π

_j

₀

_,k

_≤

Z

1

0

v

π

(t)

X

k∈∆

j0

φ

2

_j

₀

_,k

(t)dt

≤ C2

j

0

kvk

π

π

=

C

′

2

j

0

_.

(20) Thus, onsidering(16),(19),(18)and(20)weobtainfor

π

≥ 1

,

S

1

≤ C2

j0π

2

_n

−

π

2

_.

(21)

Takingina ountthat

2

j

0

_{≃ n}

1+2s

1

,theinequalities(14),(15)and(21)implythat

E

_f

₍

_{k ˆ}

f

l

_{− fk}

π

_π

)

≤ C(2

j0π

2

n

−

π

2

+ 2

−j

0

sπ

)

≤ C

′

n

−α

1

π

_.

Considering(13),wededu ethatfor

π

≥ p ≥ 1

,

sup

f

∈B

s,π,r

(L)

E

_f

₍

_{k ˆ}

_f

l

_{− fk}

p

p

)

≤ Cn

−α

1

p

.

This ompletestheproofofTheorem3.3.

3.1.2 LowerBound

Now, introdu eakeytheoremwhi h will beintensivelyused to exhibitthe lowerbounds overgeneral Besovballs.

Firstofall,remarkthatforany

j

and

N

of

N

∗

,thereexistsa onstant

2N

− 2 ≥ C

∗

≥ 2

su hthat

r

_∗

=

2

j

_{− C}

∗

2N

− 1

∈ N

∗

(22)

Theorem3.4. Let

j

anintegerdependingon

n

,

(ω

j,k

)

k

∈R

j

axedsequen eand

ε

asequen esu hthat

ε

= (ε

k

)

k

∈R

j

∈ {−1, 1}

r

∗

where

R

j

=

{2

j

− l(2N − 1) + N − 2; l = 1, 2, ..., r

∗

}.

(23) and

r

∗

isdenedby (22). Let ussetthe fun tions

g

ε

(x) = γ

j

X

k

∈R

j

ω

j,k

ε

k

ψ

j,k

(T (x))

(24)

where

γ

j

is hosen in su haway that

g

ǫ

belongsto

B

T

s,π,r

(L)

. Forsu h

ǫ

, ifwe onsider

ε

∗

k

= (ε

′

i

)

i∈R

j

denedby

ε

′

_i

= ε

i

1

{i6=k}

− ε

i

1

{i=k}

,

thenfor any estimator

f

ˆ

wehave

U

j

=

sup

g

ǫ

∈B

T

s,π,r

(L)

E

_g

_ǫ

₍

_{k ˆ}

f

_{− g}

ǫ

k

p

p

)

≥

e

−λ

₂

γ

_j

p

X

k

∈R

j

ω

_j,k

p

inf

ε

i

∈{−1,+1}

i6=k

P

_g

_ǫ

₍

_∧

_n

_(g

_ε

∗

k

, g

ǫ

) > e

−λ

₎

_kψ

j,k

k

p

w,p

where

∧

n

(g

ε

∗

k

, g

ǫ

)

denotes thelikelihoodratiobetween thelaws indu edby

g

ε

∗

k

and

g

ε

k

denedby

∧

n

(g

ε

∗

k

, g

ǫ

) =

dP

g

ε∗

_k

dP

g

ε

.

(25)

Proof of Theorem 3.4. Sin e

T

is in reasingwith

T

(0) = 0

and

T

(1) = 1

, forall

k

belongingto

R

j

we havethe

S

T

j,k

'sdened by

S

_j,k

T

= Supp(ψ

j,k

(T (.))) = [T

−1

(

k

_{− N + 1}

2

j

), T

−1

(

k

+ N

2

j

)]

whi hsatisfy

S

T

_j,k

_{∩ S}

_j,k

T

′

=

∅ for k 6= k

′

, k, k

′

∈ R

j

and

∪

k

∈R

j

S

_j,k

T

= [T

−1

(

C

∗

− 1

2

j

), T

−1

(1

−

1

2

j

)]

⊂ [0, 1].

In the ase where

T

= I

d

, weadopt the notation

S

j,k

instead of

S

T

j,k

. Denote by

G

the set of all

g

ε

denedby(24). Foranyestimator

f

ˆ

,let

W

_j,k

1

=

Z

S

T

j,k

| ˆ

f

(x)

− γ

j

ε

k

ω

j,k

ψ

j,k

(T (x))

|

p

dx

and

W

_j,k

2

=

Z

S

T

j,k

| ˆ

f

(x) + γ

j

ε

k

ω

j,k

ψ

j,k

(T (x))

|

p

dx.

Usingthefa tthat the

S

T

j,k

aredisjoint,foranypositivesequen e

(δ

j,k

)

k

∈R

j

,wehave

U

j

≥

1

card(

_G)

X

ε

E

_g

_ε

₍

_{k ˆ}

f

_{− g}

ε

k

p

p

)

≥

_card(

1

G)

X

k∈R

j

X

ε

E

_g

_ε

₍

Z

S

T

j,k

| ˆ

f

(x)

− γ

j

ǫ

k

ω

j,k

ψ

j,k

(T (x))

|

p

dx)

(26) Bythedenition of

ǫ

∗

k

andthefa tthat forall

k

∈ R

j

Card(

_{G) = 2Card(ǫ, ǫ}

i

∈ {−1, +1}, i 6= k, i, k ∈ R

j

),

weobtain

U

j

≥

1

card(

_G)

X

k

∈R

j

X

ε

i

∈{−1,+1}

i6=k

E

_g

_ε

_(W

_j,k

1

₊

_∧

_n

_(g

_ε

∗

k

, g

ǫ

)W

2

j,k

)

≥

1

₂

X

k∈R

j

inf

ε

i

∈{−1,+1}

i6=k

δ

p

_j,k

E

_g

ε

(1

{W

_j,k

1

≥δ

p

_j,k

}

+ e

−λ

₁

{∧

n

(g

ε∗

_k

,g

ǫ

)>e

−λ

}

1

{W

_j,k

2

≥δ

p

_j,k

}

)

≥

e

−λ

₂

X

k

∈R

j

inf

ε

i

∈{−1,+1}

i6=k

δ

_j,k

p

E

_g

ε

(1

{∧

n

(g

ε∗

_k

,g

ǫ

)>e

−λ

}

(1

{W

j,k

2

≥δ

p

j,k

}

+ 1

{W

1

j,k

≥δ

p

j,k

}

)).

(27)

Now, onsiderthesequen e

δ

j,k

denedby

δ

j,k

= γ

j

ω

j,k

kψ

j,k

k

w,p

.

UsingMinkowski'sinequalityandthe hangeofvariable

y

= T (x)

,weseethat

(W

1

j,k

)

1

p

+ (W

2

j,k

)

1

p

≥ 2γ

j

ω

j,k

kψ

j,k

(T (.))

k

p

= 2δ

j,k

.

Therefore

1

_{W

2

j,k

≥δ

p

j,k

}

≥ 1

{W

1

j,k

≤δ

p

j,k

}

.

(28) Putting(26),(27)and(28)together,wededu ethat

U

j

≥

e

−λ

2

γ

p

j

X

k

∈R

j

ω

p

_j,k

inf

ε

i

∈{−1,+1}

i6=k

P

_g

ǫ

(

∧

n

(g

ε

∗

k

, g

ǫ

) > e

−λ

₎

_kψ

j,k

k

p

w,p

.

Theorem3.4 anbeviewedasageneralizationofLemma

10.2

whi happearedinthebookHärdle, Kerkya harian,Pi ardandTsybakov(1998).

Theorem3.5. Let

∞ > p ≥ 1

. There existsa onstant

c >

0

su hthatfor

N > s >

0

,

∞ ≥ π ≥ 1

and

∞ ≥ r ≥ 1

wehave

inf

ˆ

f

sup

f

∈B

s,π,r

(L)

E

_f

₍

_{k ˆ}

_f

_{− fk}

p

p

)

≥ cn

−α

1

p

.

Proof of Theorem 3.5. Let

j

anintegerto be hosen below. Considerthefun tions

g

ǫ

dened by(24) with

ω

j,k

= 1 and T = I

d

.

Sin ethewavelet oe ientsof

g

ε

(denotedbelowby

β

∗

j,k

)areequalto

γ

j

ε

k

and

Card(R

j

) = r

∗

≤ C2

j

, wehave

2

j(s+

1

2

)

(

X

k∈R

j

|β

j,k

∗

|

π

2

−j

)

1

π

= γ

j

2

j(s+

1

2

)

(

X

k∈R

j

2

−j

)

π

1

≤ Cγ

j

2

j(s+

1

2

)

_.

Using Property 2.4 with

q(w) = 0

and taking

j

large enough, only the following onstraint on

γ

j

is ne essarytoguaranteethat

g

ε

∈ B

s,π,r

(L)

:

γ

j

≤ C

′

2

−j(s+

1

2

)

where

C

′

denotesa onstantsuitably hosen. Now, onsiderthefollowinglemmawhi hwill beproved inAppendix.

Lemma3.2. If we hose

γ

j

= n

−

1

2

then thereexist

λ >

0

and

p

0

>

0

notdependingon

n

su hthat

X

k

∈R

j

inf

ε

i

∈{−1,+1}

i6=k

P

_g

ǫ

(

∧

n

(g

ε

∗

k

, g

ǫ

) > e

−λ

₎

_{≥ p}

0

2

j

.

It followsfromLemma3.2andTheorem 3.4that:

sup

g

ε

∈B

s,π,r

(L)

E

_g

ǫ

(

k ˆ

f

− g

ε

k

p

p

)

≥

e

−λ

2

γ

p

j

X

k

∈R

j

inf

ε

i

∈{−1,+1}

i6=k

P

_g

ǫ

(

∧

n

(g

ε

∗

k

, g

ǫ

) > e

−λ

₎

_kψ

j,k

k

p

p

≥

e

−λ

₂

2

jp

2

kψk

p

p

(

1

√

_n

)

p

p

0

.

Choosing

j

su hthat

γ

j

= n

−

1

2

≃ 2

−j(s+

1

2

)

(i.e

2

j

_{≃ n}

1+2s

1

),onegets

inf

ˆ

f

sup

f

∈B

s,π,r

(L)

E

_f

₍

_{k ˆ}

_f

_{− fk}

p

_p

₎

_≥

e

−λ

2

kψk

p

p

(

2

j

2

√

n

)

p

_p

0

≥ c

′′

n

−α

1

p

_.

ThisendstheproofofTheorem3.5.

CombiningTheorem 3.3and Theorem3.5,weobtainTheorem3.1.

Remark3.2. If

v

isapositive onstant thenweobtain theusualminimaxresult. Example3.1. Let

∞ > π ≥ p ≥ 1

. Consider the model(1)with the operator

H

v

1

where

v

1

(t) = t

−

σ

2

for

− 1 < σ <

2

π

′

.

Itis learthat the ondition(10) holds. Sowe anapplyTheorem3.1.

Example3.2. Let

∞ > π ≥ p ≥ 1

. Consider the model(1)with the operator

H

v

2

where

v

2

(t) = (1

− t)

α

t

−β

for 0 < α <

1

2

and 0 < β <

1

π

′

.

Remarkthat

v

2

isnotboundedfromabove andbelow andthatthe ondition(10) holds. Sowe anapply Theorem 3.1.

Thefollowingsubse tionproposestoinvestigatetheminimaxrateover

B

s,π,r

(L)

under

L

p

lossfor othervaluesoftheparameters

(s, π, r)

andotherassumptionson

v

.

3.2 Proof of Theorem 3.2: upper bounds and lower bound 3.2.1 Upper bounds

Letus onsiderthefollowingthresholdwaveletpro edure:

ˆ

f

@

(x) =

X

k

∈∆

j1

ˆ

α

j

1

,k

φ

j

1

,k

(x) +

X

j

∈Λ

X

k

∈∆

j

ˆ

β

j,k

1

_{{| ˆ}

_β

j,k

|≥κ

√

j

n

}

ψ

j,k

(x)

(29)

wherewehaveset

ˆ

α

j,k

=

Z

1

0

φ

j,k

(t)v(t)dY

t

,

β

ˆ

j,k

=

Z

1

0

ψ

j,k

(t)v(t)dY

t

,

Λ =

_{{j ≥ τ − 1; j}

1

< j < j

2

}

for

j

1

and

j

2

theintegersverifying

2

j

1

_{≈ (n(ln(n))}

p−π

π

1

{ǫ≥0}

₎

1−2α

_{and 2}

j

2

_{≈ (n(ln(n))}

−1

{ǫ≤0}

₎

α

(s− 1

π

+ 1

p

)

(30) where

α

= max(α

1

, α

2

).

Followingstepby step theproof of Theorem 3whi h appeared in Donoho,Johnstone, Kerkya harian andPi ard(1996),one anshowthattheestimator(29)attainstheupperbounddes ribein Theorem 3.2. Thisisanimmediate onsequen eofthefollowinglemma:

Lemma3.3. Assumethat the ondition(11) holds. Then forany

κ

thereexist

C >

0

and

κ

′

satisfying

• E

f

(

| ˆ

β

j,k

− β

j,k

|

2p

)

≤ Cn

−p

• P

f

(

| ˆ

β

j,k

− β

j,k

| ≥

κ

′

2

q

j

n

)

≤ C2

−κj

.

Theproofisreje tedin Appendix.

3.2.2 Lowerbound

Theorem 3.6. Let

∞ > p ≥ 1

. There exists a onstant

c >

0

su h that for

N > s >

1

π

,

0

≥ ǫ

and

∞ ≥ r ≥ 1

wehave

inf

ˆ

f

sup

f∈B

s,π,r

(L)

E

_f

₍

_{k ˆ}

f

_{− fk}

p

_p

)

_{≥ c(}

ln(n)

n

)

α

2

p

_.

Proof of Theorem 3.6. Considerthefollowingfamily

{g

0

= 0; g

k

= γ

j

ψ

j,k

, k

∈ R

j

}

where

R

j

is dened by(23). In order toproveTheorem 3.6, letus introdu eatheorem whi h anbe viewasanadaptedversionofLemma

10.1

ofHärdle,Kerkya harian,Pi ardandTsybakov(1998). Theorem3.7. Assumethe following onditions arefullled:

• ∀k ∈ R

j

,

γ

j

is hosen su hthat

g

k

∈ B

s,π,r

(L),

•

Thereexistsa onstant

p

0

>

0

satisfying

X

k

∈R

j

P

_g

_k

_(Λ(g

₀

, g

k

)

≥ 2

−λ

∗

_j

)

≥ p

0

2

j

(31) for axed

λ

∗

su hthat

1

≥ λ

∗

_>

₀

. Then for anyestimator

f

ˆ

wehave

sup

f∈B

s,π,r

(L)

E

_f

₍

_{k ˆ}

f

_{− fk}

p

_p

)

≥ 2

−p

₂

j(

p

2

−1)

γ

p

j

kψk

p

p

p

0

2

.

Proof of Theorem 3.7. Forsakeofsimpli ity,letusdenoteby

d

the

L

p

metri ,i.e. forany

f

and

g

whi h belongto

L

p

_{([0, 1])}

d(f, g) = (

Z

1

0

|f(t) − g(t)|

p

_dt)

1

p

.

Put

δ

j

=

γ

j

2

2

j(

1

2

−

1

p

)

kψk

p

.

(32)

FromCheby hev'sinequality,weseethat

δ

_j

−p

sup

f

∈B

s,π,r

(L)

E

_f

₍

_{k ˆ}

_f

_{− fk}

p

p

)

≥ δ

j

−p

sup

k

∈R

j

∪{0}

E

_g

k

(

k ˆ

f

− g

k

k

p

p

)

≥

sup

k

∈R

j

∪{0}

P

_g

k

(

k ˆ

f

− g

k

k

p

≥ δ

j

)

≥ max(

_r

1

∗

X

k

∈R

j

P

_g

_k

_{(d( ˆ}

f , g

k

)

≥ δ

j

), P

g

0

(d( ˆ

f , g

0

)

≥ δ

j

)).

(33) Sin e

r

∗

≤ 2

j

itsu estoprovethat

max(2

−j

X

k

∈R

j

P

_g

_k

_{(d( ˆ}

f , g

k

)

≥ δ

j

), P

g

0

(d( ˆ

f , g

0

)

≥ δ

j

))

≥

p

0

2

.

(34)

Assumeonthe ontrarythat(34)isfalse. Thenthereexists anestimator,say

f

∗

,su h that

max(2

−j

X

k

∈R

j

P

_g

k

(d(f

∗

, g

k

)

≥ δ

j

), P

g

0

(d(f

∗

, g

0

)

≥ δ

j

)) <

p

0

2

.

Inparti ular,wehave

P

_g

₀

_(d(f

∗

_{, g}

₀

₎

_{≥ δ}

_j

_{) <}

p

0

2

(35)

andsin ethereexists

c >

0

su h that

r

∗

≥ c2

j

(forinstan e,

c

=

1

N

(2N −1)

),wehave

2

−j

X

k

∈R

j

P

_g

k

(d(f

∗

_{, g}

k

) < δ

j

) > c

−

p

0

2

.

(36)

Putting(31)and(36)together,weobtainthatforany

k

∈ R

j

X

k

∈R

j

P

_g

_k

₍

_{d(f

∗

, g

k

) < δ

j

} ∩ {Λ(g

0

, g

k

)

≥ 2

−λ

∗

_j

}) ≥

X

k

∈R

j

P

_g

_k

_(d(f

∗

, g

k

) < δ

j

)

+

X

k

∈R

j

P

_g

_k

_(Λ(g

₀

, g

k

)

≥ 2

−λ

∗

_j

)

− (2N)

−1

₂

j

>

(c

−

p

₂

0

)2

j

+ p

0

2

j

− c2

j

>

p

0

2

2

j

_.

(37)

Wenowusethe

δ

j

dened in(32). Firstforall

k

∈ R

j

d(g

k

, g

0

) = γ

j

kψ

j,k

k

p

= 2δ

j

andthetriangularinequalityimpliesthat

∪

k

∈R

j

{d(f

∗

_{, g}

k

) < δ

j

} ⊂ {d(f

∗

, g

0

)

≥ δ

j

}.

Se ondforall

k

6= k

′

_{∈ R}

j

wehave

d(g

k

, g

′

k

) =

γ

j

(

kψ

j,k

k

p

+

kψ

j,k

′

k

_p

)

≥ γ

j

kψ

j,k

k

p

Consequentlytheevents

{d(f

∗

_{, g}

k

) < δ

j

}

aredisjointfor

k

6= k

′

_{∈ R}

j

. Itfollowsfrom (37)that

P

_g

0

(d(f

∗

, g

0

)

≥ δ

j

)

≥ P

g

0

(

∪

k

∈R

j

d(f

∗

_{, g}

k

) < δ

j

))

=

X

k

∈R

j

P

_g

₀

_(d(f

∗

_{, g}

_k

_{) < δ}

_j

₎₎

=

X

k

∈R

j

E

_g

k

(Λ(g

0

, g

k

)1

{d(f

∗

,g

k

)<δ

j

}

)

≥ 2

−λ

∗

j

X

k

∈R

j

E

_g

k

(1

{d(f

∗

,g

k

)<δ

j

}

1

{Λ(g

0

,g

k

)≥2

−λ∗j

}

)

= 2

−λ

∗

j

X

k

∈R

j

P

_g

_k

₍

_{d(f

∗

, g

k

) < δ

j

} ∩ {Λ(g

0

, g

k

)

≥ 2

−λ

∗

_j

})

>

p

0

2

2

(1−λ

∗

)j

_.

Thenwe ontradi t(35). So, ombining(33)and(34),wededu ethat foranyestimator

f

ˆ

wehave

sup

f∈B

s,π,r

(L)

E

_f

₍

_{k ˆ}

f

_{− fk}

p

_p

)

_{≥ 2}

−p

2

j(

p

2

−1)

γ

p

j

kψk

p

p

p

0

2

.

(38)

Sin ethewavelet oe ientsof

g

k

, denotedbelowby

β

∗

j,k

,areequalto

γ

j

with

k

xed, wehave

2

j(s+

1

2

)

(

|β

∗

j,k

|

π

2

−j

)

1

π

= γ

_j

2

j(s+

1

2

)

(2

−j

)

1

π

= γ

j

2

j(s+

1

2

−

1

π

)

.

Sotaking

γ

j

≤ C

′

2

−j(s+

1

2

−

1

π

)

where

C

′

denotes a onstantsuitably hosen,Property2.4with

q(w) = 0

impliesthatthe

g

k

belong to

B

s,π,r

(L)

. Now onsiderthefollowinglemma:

Lemma3.4. Let

γ

j

= c

0

q

ln(n)

n

. Ifthere existsa onstant

c >

0

su hthat for

n

largeenough

ln(2

j

₎

≥ c ln(n)

(39)

thenfor axed

1

≥ λ

∗

_>

₀

anda

c

0

smallenough thereexistsa onstant

p

0

>

0

satisfying

X

k

∈R

j

P

_g

_k

_(Λ(g

₀

, g

k

)

≥ 2

−λ

∗

_j

)

≥ p

0

2

j

.

(40) Thus, hoosing

γ

j

= c

0

r

ln(n)

n

,

i.e 2

j

≃ (

r

_n

ln(n)

)

1

s+ 1

₂

− 1

_π

andremarkingthatfor

n

largeenoughwehave

ln(2

j

₎

≥

_{2(s +}

1

1

2

−

1

π

)

(ln(n)

_{− ln(ln(n))) + ln(c)}

≥

_{4(s +}

1

1

2

−

1

π

)

ln(n)

≥

_{4N + 2}

1

ln(n),

the ondition(39)andafortiori,the ondition(31)aresatised. SoTheorem3.7impliesthat

sup

f∈B

s,π,r

(L)

E

_f

₍

_{k ˆ}

f

_{− fk}

p

_p

)

≥ 2

−p

₂

j(

p

2

−1)

γ

p

j

kψk

p

p

p

0

2

≥ c(

ln(n)

_n

)

p

2

(

r

_n

ln(n)

)

p

2

−1

s+ 1

₂

− 1

π

= c(

ln(n)

n

)

α

2

p

_.

PuttingSubse tion'Upperbounds'andTheorem3.6together,weestablishTheorem3.2.

3.3 When

v

does not belong to

L

π

′

([0, 1])

Thelastpartofthis se tionis enteraroundthefollowingquestion:

Question 3.1. Let

π

≥ p > 2

. Can we extend the minimaxproperties without deterioratingthe rate of onvergen e over

B

s,π,r

(L)

for other

v

i.e fun tions whi h does not satisfy the assumption '

v

belongs to

L

π

_{([0, 1])}

'(seeTheorem3.1)?

The following theorem gives a beginning of answer by onsidering a fun tion

v

whi h does not belongto

L

p

_{([0, 1])}

and, afortiori,to

L

π

_{([0, 1])}

.

Theorem 3.8. Let

∞ > p > 2

and

∞ ≥ π ≥ p

. Assumethat we observe model (1)with the operator

H

v

∗

where

v

_∗

(t) = t

−

σ

2

for

2

p

< σ <

1.

(41)

Then for

N > s >

0

and

∞ ≥ r ≥ 1

thereexisttwo onstant

C >

0

and

c >

0

su hthat

cn

− ˜

αp

_{≤ inf}

ˆ

f

sup

f∈B

s,π,r

(L)

E

_f

₍

_{k ˆ}

f

_{− fk}

p

_p

)

≤ Cn

−α

′

_p

where

α

′

=

s

2s + 1 + σ

₋

2

_π

and

˜

α

=







α

1

_π(σp−2)

π

−p

> s >

0,

π > p,

s+

1

p

−

1

π

2s+1+σ−

2

π

N > s

_≥

_π

_(σp−2)

π

−p

,

π

_{≥ p.}

Proof of Theorem 3.8. First,introdu ethefollowinglemma whi hwill beprovedinAppendix.

Lemma3.5. Let

π >

2

. Letus onsider

ρ

j,k

denedby(17) with

v

= v

∗

(see(41)) and

η

j,k

denedby

η

j,k

=

s

Z

1

0

1

v

2

∗

(t)

ψ

2

j,k

(t)dt.

(42)

Then thereexisttwo onstant

C >

0

and

c >

0

su hthat

c2

jσπ

2

≤

X

k

∈R

j

η

_j,k

−π

_≤

X

k

∈∆

j

ρ

π

_j,k

_{≤ C2}

jσπ

2

.

(43) 3.3.1 Upper bound

Letus onsiderthelinearestimator

f

ˆ

l

denedin(12)where

v

isdenedby(41). Puttingtheinequalities (18)and (43)together,onegets

sup

f

∈B

s,π,r

(L)

E

_f

₍

_{k ˆ}

_f

l

_{− fk}

p

_p

₎

_{≤ C(2}

j

0

(

π

2

−1)

_n

−

π

2

X

k∈∆

j0

ρ

π

j,k

+ 2

−j

0

sπ

)

p

π

≤ C

′

₍₂

j

0

(

π

₂

−1+

σπ

₂

)

_n

−

π

₂

_{+ 2}

−j

0

sπ

₎

p

_π

≤ C

′′

n

−α

′

p

for

j

0

theintegersatisfying

2

j

0

_{≃ n}

1

1+2s+σ− 2

_π

3.3.2 Lowerbound

Considerthefun tions

g

ǫ

introdu edin(24)with

ω

j,k

= η

j,k

−1

and T = I

d

where

η

j,k

is dened by(42). Sin e thewavelet oe ientsof

g

ε

, denoted belowby

β

∗

j,k

, are equalto

η

_j,k

−1

γ

j

ε

k

, theinequality(43)givesus

2

j(s+

1

2

)

(

X

k

∈R

j

|β

j,k

∗

|

π

2

−j

)

1

π

=

_γ

j

2

j(s+

1

2

)

(

X

k

∈R

j

η

_j,k

−π

2

−j

)

π

1

≤ Cγ

j

2

j(s+

1

2

+

σ

2

−

π

1

)

.

Using Property 2.4 with

q(w) = 0

and taking

j

large enough, only the following onstraint on

γ

j

is ne essarytoguaranteethat

g

ε

∈ B

s,π,r

(L)

:

γ

j

≤ C

′

2

−j(s+

1

2

+

σ

2

−

1

π

)

where

C

′

denotesa onstantsuitably hosen. Now, onsiderthefollowingLemmawhi hwillbeproved inAppendix.

Lemma3.6. If we hose

γ

j

= n

−

1

2

then thereexist

λ >

0

and

p

0

>

0

notdependingon

n

su hthat

inf

ε

i

∈{−1,+1}

i6=k

P

_g

_ǫ

₍

_∧

_n

_(g

_ε

∗

k

, g

ǫ

) > e

−λ

₎

_{≥ p}

0

,

∀k ∈ R

j

.

PuttingLemma 3.6andTheorem3.4together,weobtain

sup

g

ε

∈B

s,π,r

(L)

E

_g

_ǫ

₍

_{k ˆ}

f

_{− g}

ε

k

p

p

)

≥

e

−λ

2

γ

p

j

X

k

∈R

j

inf

ε

i

∈{−1,+1}

i6=k

ω

_j,k

p

P

_g

_ǫ

₍

_∧

_n

_(g

_ε

∗

k

, g

ǫ

) > e

−λ

₎

_kψ

j,k

k

p

p

≥

e

−λ

₂

2

j(

p

2

−1)

p

₀

(

√

1

n

)

p

kψk

p

p

X

k∈R

j

η

_j,k

−p

.

Usingtheinequality(43)andthefa t that

n

1

2s+1+σ− 2

_π

≃ 2

j

,one gets

inf

ˆ

f

sup

f∈B

s,π,r

(L)

E

_f

₍

_{k ˆ}

f

_{− fk}

p

_p

)

≥ c(n

−

1

2

2

j(

1

2

+

σ

2

−

1

p

)

)

p

≥ c

′

_n

− ˜

αp

_.

ThisendstheproofofTheorem3.8.

Remark 3.3. Let

p >

2

. Considerthe fun tion

v

∗

denedby(41). It learthat for

N > s >

0

,

π

= p

and

r

≥ 1

wehave

inf

ˆ

f

sup

f

∈B

s,p,r

(L)

E

_f

₍

_{k ˆ}

_f

_{− fk}

p

p

)

≍ n

−α

∗

p

where

α

∗

=

s

1 + 2s + σ

−

2

p

.

Sowehaveprovethatif

v

doesnotbelongto

L

π

_{([0, 1])}

for

π

≥ p

thenthemininaxrateoverusual Besovballsunder

L

p

risk anbeslowerthan

n

−α

1

p

. Inparti ular,Theorem 3.8showsthatthisrateof onvergen e antrulydependonthenature of

v

.

This arisesanewquestion:

Question3.2. Canwendfun tionspa esoverwhi htheminimaxrateunderthe

L

p

riskstay'stable'for othersfun tions

v

i.e whi hdoesnotbelongto

L

π

_{([0, 1])}

? Theanswerisdevelopedinthefollowingse tion.