Nonparametric estimation of the derivatives of the stationary density for stationary processes

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Nonparametric estimation of the derivatives of the

stationary density for stationary processes

Emeline Schmisser

To cite this version:

Emeline Schmisser.

Nonparametric estimation of the derivatives of the stationary density for

stationary processes.

ESAIM: Probability and Statistics, EDP Sciences, 2013, 17, pp33-69.

�10.1080/02331888.2011.591931�. �hal-00507025�

stationary density for stationary pro esses

Emeline S hmisser

Université ParisDes artes

Laboratoire MAP5 Emeline.S hmissermath-info.univ-paris5.fr

Abstra t

Inthisarti le, ouraim isto estimatethe su essivederivativesofthestationary density

f

ofastri tlystationaryand

β

-mixingpro ess

(X

t

)

t≥0

. Thispro essisobservedatdis rete times

t

= 0, ∆, . . . , n∆

. Thesamplinginterval

∆

anbe xedorsmall. Weusea penalized least-square approa hto omputeadaptive estimators. Ifthe derivative

f

(j)

belongsto the

Besovspa e

B

α

2,∞

,thenourestimator onvergesatrate

(n∆)

−α/(2α+2j+1)

.Thenwe onsidera

diusionwithknowndiusion oe ient. Weusetheparti ularformofthestationarydensity

to omputean adaptive estimator of its rstderivative

f

′

. Whenthe samplinginterval

∆

tendsto0,andwhenthediusion oe ientisknown,the onvergen erateofourestimatoris

(n∆)

−α/(2α+1)

. Whenthediusion oe ientisknown,wealso onstru taquotientestimator ofthedrift forlow-frequen ydata.

Keywords: derivativesof thestationary density, diusion pro esses,mixing pro esses,

nonpara-metri estimation,stationarypro esses

AMSClassi ation: 62G05,60G10

1 Introdu tion

Inthisarti le,we onsiderastri tlystationary,ergodi and

β

-mixingpro ess

(X

t

, t ≥ 0)

observed atdis retetimeswithsamplinginterval

∆

. The

j

thorderderivatives

f

(j)

(

j ≥ 0

)ofthestationary density

f

areestimatedbymodelsele tion. Adaptiveestimatorsof

f

(j)

are onstru tedthanksto

apenalizedleast-squaremethodandthe

L

2

riskoftheseestimatorsis omputed.

Numerousarti lesdealwithnonparametri estimationofthestationarydensity(orthe

deriva-tivesofthestationarydensity)forastri tlystationaryandmixingpro essobservedin ontinuous

time. Forinstan e,Bosq[4℄usesakernelestimator,ComteandMerlevède[5℄realizeaproje tion

estimationand Leblan [16℄ utilizeswavelets. Under theCastellana andLeadbetter's onditions,

when

f

belongstoaBesovspa e

B

α

2,∞

,theestimatorof

f

onvergesattheparametri rate

T

−1/2

(where

T

isthetimeof observation). Thenon parametri estimationofthestationarydensityof astationaryandmixingpro essobservedatdis retetimes

t = 0, ∆, . . . , n∆

hasalsobeenstudied, espe iallywhenthesamplinginterval

∆

isxed. Forexample,Masry[19℄ onstru tswavelets es-timators,ComteandMerlevède[ 7℄andLerasle[17℄useapenalizedleast-square ontrastmethod.

The

L

2

rateof onvergen e oftheestimatoris inthat ase

n

−α/(2α+1)

. ComteandMerlevède[5℄

demonstratethat, ifthe sampling interval

∆ → 0

, the penalized estimator of

f

onvergeswith rate

(n∆)

−α/(2α+1)

and, under the onditionsof Castellanaand Leadbetter,theparametri rate

of onvergen eisrea hed.

Therearelesspapersabouttheestimationofthederivativesofthestationarydensity,andthe

mainresultsarefor independentandidenti ally distributedrandom variables. Forinstan e,Rao

[22℄estimatesthesu essivederivatives

f

(j)

ofamulti-dimensionalpro essbyawaveletmethod.

Hebounds the

L

2

riskofhis estimatorand omputesthe rateof onvergen eonSobolevspa es.

This estimator onvergeswith rate

n

−α/(2α+2j+1)

. Hosseinioun et al. [13℄ estimate the partial

derivativesofthestationarydensityofamixingpro essbyawaveletmethod,andtheirestimators

onvergewithrate

(n∆)

−α/(2α+1+2j)

.

Classi al examples of

β

-mixing pro esses are diusions: if

(X

t

)

is solution of the sto hasti dierentialequation

dX

t

= b(X

t

)dt + σ(X

t

)dW

t

and

X

0

= η,

then,withsome lassi aladditional onditionson

b

and

σ

,

(X

t

)

isexponentially

β

-mixing.Dalalyan and Kutoyants [9℄ estimate the rst derivative of the stationary density for a diusion pro ess

observedat ontinuous time. They prove that the minimax rate of onvergen e is

T

−2α/(2α+1)

where

T

is the time of observation. This is the samerate of onvergen e asfor non parametri estimatorof

f

.

A possible appli ation is, for diusion pro esses, the estimation of the drift fun tion

b

by quotient. Indeed, when

σ = 1

, wehavethat

f

′

_{= 2bf}

. Thedrift estimationis well-knownwhen

thediusion it observedat ontinuous time orfor high-frequen y data (see Comte et al. [6℄ for

instan e), but it is far more di ult when

∆

is xed. Gobet et al. [ 12℄ build non parametri estimatorsof

b

and

σ

when

∆

isxedandprovethat theirestimatorsrea htheminimax

L

2

risk.

However,theirestimatorsarebuiltwitheigenvaluesoftheinnitesimalgeneratorandaredi ult

toimplement.

In this paper, in a rst step, we onsider a stri tly stationary and

β

-mixing pro ess

(X

t

)

t≥0

observedat dis rete times

t = 0, ∆, . . . , n∆

. The su essive derivatives

f

(j)

(

0 ≤ j ≤ k

)of the stationarydensity

f

areestimated either ona ompa tset, oron

R

thanksto apenalized least-squaremethod. Weintrodu easequen eofin reasinglinearsubspa es

(S

m

)

and,forea h

m

,we onstru tanestimatorof

f

(j)

byminimisinga ontrastfun tionover

S

m

. Then,apenaltyfun tion

pen(m)

is introdu edto sele t anestimatorof

f

(j)

in the olle tion. When

f

(j)

_{∈ B}

α

2,∞

, the

L

2

riskof thisestimator onvergeswith rate

(n∆)

−2α/(2α+2j+1)

and the pro edure doesnotrequire

theknowledgeof

α

. When

j = 0

,thisistherateof onvergen eobtainedbyComteandMerlevède [7, 5℄. Moreover, when

α

is known, Rao [22℄ obtained a rate of onvergen e

n

−2α/(2α+2j+1)

for

independentvariables.

Inase ondstep,weassumethatthepro ess

(X

t

)

issolutionofasto hasti dierentialequation ofknowndiusion oe ient

σ

. Then

f

′

anbeestimatedbyestimating

2bf

and

f

. Anestimator of

2bf

is builteither on a ompa tset,oron

R

by apenalized least-square ontrastmethod. It only onvergeswhenthesamplinginterval

∆ → 0

,butinthis ase,itsrateof onvergen eisbetter thanforthepreviousestimator: itis

(n∆)

−2α/(2α+1)

when

f

′

_{∈ B}

α

2,∞

(andnot

(n∆)

−2α/(2α+3)

).

ThisistheminimaxrateobtainedbyDalalyanandKutoyants[9℄with ontinuousobservations.

Then, an estimator by quotient of the drift fun tion

b

is onstru ted. When

∆

is xed, it rea hestheminimaxrateobtainedbyGobetet al.[ 12℄.

InSe tion2,anadaptiveestimatorofthesu essivederivatives

f

(j)

ofthestationarydensity

f

ofastationaryand

β

-mixingpro essis omputedbyapenalized leastsquaremethod. InSe tion 3,onlydiusionswithknowndiusion oe ientsare onsidered. Anadaptiveestimatorof

f

′

(in

fa t,anestimatorof

2bf

)isbuilt. InSe tion4,aquotientestimatorof

b

is onstru ted. InSe tion 5, the theoreti al resultsare illustratedvia various simulationsusing several models. Pro esses

(X

t

)

aresimulatedbytheexa tretrospe tivealgorithmof Beskosetal.[3℄ . Theproofsaregiven inSe tion6. IntheAppendix, thespa esoffun tionsareintrodu ed.

2 Estimation of the su essive derivatives of the stationary

density

2.1 Model and assumptions

Inthisse tion,astationarypro ess

(X

t

)

t≥0

isobservedatdis retetimes

t = 0, ∆, . . . , n∆

andthe su essive derivatives

f

(j)

of the stationary density

f = f

(0)

are estimated for

0 ≤ j ≤ k

. The samplinginterval

∆

isxedortendsto0. Theestimationset

A

iseithera ompa t

[a

0

, a

1

]

,or

R

. Letus onsiderthenorms

k.k

∞

= sup

A

|.|

,

k.k

L

2

= k.k

_L

2

_(A)

and

h., .i = h., .i

L

2

_(A)

.

(2.1)

Thepro ess

(X

t

)

isergodi , stri tly stationary andarithmeti ally orexponentially

β

-mixing. Apro essisarithmeti ally

β

-mixingifits

β

-mixing oe ientsatises:

β

X

(t) ≤ β

0

(1 + t)

−(1+θ)

(2.2) where

θ

and

β

0

aresomepositive onstants. Apro essisexponentially(orgeometri ally)

β

-mixing ifthereexists twopositive onstants

β

0

and

θ

su hthat:

β

X

(t) ≤ β

0

exp (−θt)

(2.3)

AssumptionM2.

The stationary density

f

is

k

times dierentiable and, for ea h

j ≤ k

,its derivatives

f

(j)

belong to

L

2

_{(A) ∩ L}

1

_(A)

. Moreover,

f

(j)

satises

R

A

x

2

_f

(j)

_(x)

2

dx < +∞

. Remark2.1.If

A = [a

0

, a

1

]

,AssumptionM2 isonly

∀j ≤ k

,

f

(j)

_{∈ L}

2

_(A)

.

Our aim is to estimate

f

(j)

by model sele tion. Therefore an in reasing sequen e of nite

dimensionallinearsubspa es

(S

m

)

isneeded. Onea hof these subspa es,anestimatorof

f

(j)

is

omputed,andthankstoapenaltyfun tiondependingon

m

,thebestpossibleestimatoris hosen. Letusdenote by

C

l

the spa eoffun tions

l

timesdierentiableon

A

and witha ontinuous

ℓ

th derivative,and

C

l

m

thesetofthepie ewisefun tions

C

l

. Toestimate

f

(j)

,

0 ≤ j ≤ k

ona ompa t set,weneedasequen eoflinearsubspa esthat satisestheassumption:

AssumptionS1: Estimationon a ompa t set. 1. Thesubspa es

S

m

arein reasing,of -nitedimension

D

m

andin ludedin

L

2

_(A)

.

2. Forany

m

,anyfun tion

t ∈ S

m

is

k

timesdierentiable(belongsto

C

k−1

_∩C

k

m

)andsatises:

∀j ≤ k,

t

(j)

(a

0

) = t

(j)

(a

1

) = 0.

3. There existsanorm onne tion: for any

j ≤ k

,thereexistsa onstant

ψ

j

su hthat:

∀m, ∀t ∈ S

m

,

t

(j)

2

∞

≤ ψ

j

D

2j+1

m

ktk

2

L

2

.

Letus onsider

(ϕ

λ,m

, λ ∈ Λ

m

)

anorthonormal basisof

S

m

with

|Λ

m

| = D

m

. Wehavethat

Ψ

2

_j,m

(x)

∞

≤ ψ

j

D

2j+1

m

where

Ψ

2

j,m

(x) =

P

λ∈Λ

m



ϕ

(j)

λ,m

(x)



2

.

4. There existsa onstant

c

su hthat,for any

m ∈ N

,any fun tion

t ∈ S

m

:

t

(j)

L

2

≤ cD

2j

m

ktk

2

L

2

.

5. Foranyfun tion

t

belonging tothe unitballof aBesovspa e

B

α

2,∞

:,

kt − t

m

k

2

L

2

≤ D

−2

_m

∨ D

−2α

_m

where

t

m

isthe orthogonal

(L

2

₎

proje tion of

t

over

S

m

.

Remark 2.2.Be ause of Point 2, theproje tion

t

m

onvergesveryslowly to

t

on theboundaries of the ompa t

A = [a

0

, a

1

]

and the inequality

kt − t

m

k

2

L

2

≤ D

−2α

_m

an notbe satisedfor any

t ∈ B

α

2,∞

.

Inthe Appendix, several sequen es oflinear subspa essatisfyingthis property aregiven. To

estimate

f

(j)

on

R

, slightly dierent assumptions are needed: let us onsider an in reasing se-quen e oflinear subspa es

S

m

generated by anorthonormalbasis

{ϕ

λ,m

, λ ∈ Z}

. We havethat

dim(S

m

) = ∞

,sotobuildestimators,weusetherestri tedspa es

S

m,N

=

Ve t

(ϕ

λ,m

, λ ∈ Λ

m,N

)

with

|Λ

m,N

| < +∞

. Thefollowingassumptioninvolvesthesequen esoflinearsubspa es

(S

m

)

and

AssumptionS2: Estimationon

R

. 1. The sequen eof linearsubspa es

(S

m

)

isin reasing. 2. Wehavethat

|Λ

m,N

| := dim(S

m,N

) = 2

m+1

_{N + 1}

. 3.

∀m, N ∈ N, ∀t ∈ S

m,N

: t ∈ C

k−1

_{∩ C}

k

m

and

∀j < k

,

lim

|x|→∞

t

(j)

_{(x) = 0}

. 4.

∃ψ

j

∈ R

+

,

∀m ∈ N

,

∀t ∈ S

m

,

∀j ≤ k

,

t

(j)

2

∞

≤ ψ

j

2

(2j+1)m

_ktk

2

L

2

.

Parti ularly,

Ψ

2

m

(x)

2

∞

=

X

λ∈Z



ϕ

(j)

_λ,m

(x)



2

2

∞

≤ ψ

j

2

(2j+1)m

.

5.

∃c

,

∀m ∈ N

,

∀t ∈ S

m

,

∀j ≤ k

:

t

(j)

2

L

2

≤ c2

2jm

_ktk

2

L

2

.

6. Foranyfun tion

t ∈ L

2

_{∩ L}

1

_(R)

su h that

R x

2

_t

2

_{(x)dx < +∞}

,

kt

m

− t

m,N

k

2

_L

2

≤ c

2

m

N

where

t

m

isthe orthogonal (

L

2

) proje tion of

t

over

S

m

and

t

m,N

itsproje tion over

S

m,N

. 7. There exists

r ≥ 1

su h that, for any fun tion

t

belonging to the unit ball of a Besov spa e

B

α

2,∞

(with

α < r

),

kt − t

m

k

2

L

2

≤ 2

−2mα

.

Proposition 2.1.

Ifthefun tion

ϕ

generatesa

r

-regularmultiresolutionanalysisof

L

2

,with

r ≥ k

,thenthesubspa es

S

m

=

Ve t

{ϕ

λ,m

, λ ∈ Z}

and

S

m,N

=

Ve t

{ϕ

λ,m

, λ ∈ Λ

m,N

}

(where

ϕ

λ,m

(x) = 2

m/2

_{ϕ (2}

m

_{x − λ)}

and

Λ

m,N

= {λ ∈ Z, |λ| ≤ 2

m

_{N }}

) satisfyS2.

Forthedenitionofthemulti-resolutionanalysis,see Meyer[20℄, hapter2.

2.2 Risk of the estimator for xed

m

Anestimator

ˆ

g

j,m

of

g

j

:= f

(j)

is omputedbyminimising the ontrastfun tion

γ

j,n

(t) = ktk

2

L

2

−

2(−1)

j

n

n

X

k=1

t

(j)

(X

k∆

).

UnderAssumptionsS1orS2:

E

_(γ

_j,n

_{(t)) = ktk}

2

L

2

−2 (−1)

j

D

t

(j)

_{, f}

E

_{= ktk}

2

L

2

−2

D

t, f

(j)

E

₌

t − f

(j)

2

L

2

−C

where

C =

f

(j)

2

L

2

.

IfAssumptionS1issatised,letusdenote

ˆ

g

j,m

(t) = arg inf

t∈S

m

γ

j,n

(t),

and,underAssumptionS2,

ˆ

g

j,m,N

(t) = arg inf

t∈S

m,N

γ

j,n

(t).

Wehavethetwofollowingtheorems:

Theorem2.1: Estimationon a ompa t set.

UnderAssumptionsM1-M2 andS1,the estimator risksatises, forany

j ≤ k

and

m ∈ N

:

E



kˆg

j,m

− g

j

k

2

_L

2



≤ kg

j,m

− g

j

k

2

_L

2

+ 8cβ

0

ψ

j

D

2j+1

m

n



1 ∨

_θ∆

1



where

g

j,m

istheorthogonal

(L

2

₎

proje tion of

g

j

over

S

m

. The onstants

β

0

and

θ

aredenedin (2.2)or (2.3),

ψ

j

isdenedinAssumptionS1and

c

isauniversal onstant.

Theorem2.2: Estimationon

R

.

UnderAssumptionsM1-M2 andS2,for any

j ≤ k

and

m ∈ N

:

E



_kˆg

_j,m,N

_{− g}

_j

_k

2

L

2



≤ kg

j,m

− g

j

k

2

_L

2

+ C

2

m

N

+ 8cβ

0

ψ

j

2

(2j+1)m

n



1 ∨

_θ∆

1



where

C

depends on

R

∞

−∞

x

2

_g

2

_(x)dx

and of the hosen sequen e of linear subspa es

(S

m,N

)

. A - ording toAssumptionS26.,if

N ≥ (n ∧ nθ∆)

,

E



_kˆg

_j,m,N

_{− g}

_j

_k

2

L

2



≤ kg

j,m

− gk

2

_L

2

+ cβ

0

2

(2j+1)m

n



1 ∨

1

θ∆



.

If the random variables

(X

0

, . . . , X

n

)

are independent, the derivatives of the density an be estimatedinthesamewayandthetwoprevioustheorems(aswellasthetheoremsfortheadaptive

risk) anbeappliedifweset

θ = +∞

.

When

∆ = 1

,theriskboundisthesameasinHosseiniounetal.[13℄.

2.3 Optimisation of the hoi e of

m

Under Assumption S1 and if

g

j

belongs to the unit ball of a Besov spa e

B

α

2,∞

with

α ≥

1

, then

kg

j,m

− g

j

k

2

L

2

≤ cD

_m

−2

and the best bias-varian e ompromise is obtained for

D

m

∼

(n (1 ∨ θ∆))

1/(2j+3)

_.

Inthat ase,

E



_kˆg

_j,m

_{− g}

_j

_k

2

L

2



≤ (n ∨ nθ∆)

−2/(2j+3)

.

IfAssumptionS2issatisedandif

g

j

belongsto

B

α

2,∞

,with

r ≥ α

,then

kg

j,m

− g

j

k

2

L

2

≤ c2

−2mα

.

If

N ≥ n (1 ∧ θ∆)

,thebestbias-varian e ompromiseisobtainedfor

m ∼

1

2j + 1 + 2α

log

2

(n (1 ∨ θ∆))

andthen

E



kˆg

j,m,N

− g

j

k

2

_L

2



≤ (n ∨ n∆)

−2α/(2α+2j+1)

.

Rao[22℄ builds estimatorsofthe su essivederivatives

f

(j)

forindependentvariables. This

esti-mators onvergewithrate

n

−2α/(2α+2j+1)

.

2.4 Risk of the adaptive estimator on a ompa t set

Anadditionalassumptionforthepro ess

(X

t

)

isneeded: AssumptionM3.

Ifthe pro ess

(X

t

)

t≥0

isarithmeti ally

β

-mixing, thenthe onstant

θ

denedin (2.2)issu hthat

θ > 3

.

Letusset

M

j,n

= {m, D

m

≤ D

j,n

}

where

D

j,n

≤ (n∆ ∧ n)

1/(2j+2)

isthemaximal dimension.

Forany

m ∈ M

j,n

, anestimator

g

ˆ

j,m

∈ S

m

of

g

j

= f

(j)

is omputed. Letusintrodu eapenalty

fun tion

pen

j

(m)

dependingon

D

m

and

n

:

pen

j

(m) ≥ κβ

0

ψ

j

D

2j+1

m

n



1 ∨

_θ∆

1



.

Thenwe onstru tanadaptiveestimator: hoose

m

ˆ

j

su hthat

˜

g

j

:= ˆ

g

j, ˆ

m

j

where

m

ˆ

j

= arg min

m∈M

j,n

[γ

j,n

(ˆ

g

j,m

) + pen

j

(m)] .

Theorem2.3: Adaptive estimationon a ompa t set.

Thereexists auniversal onstant

κ

su hthat,if AssumptionsM1-3andS1aresatised:

E



_k˜g

_j

_{− g}

_j

_k

2

L

2



≤ C

inf

m∈M

j,n



kg

j,m

− g

j

k

2

_L

2

+ pen

j

(m)



+

c

n



1 ∨

_∆

1



andMerlevède[5℄obtainsimilarresultswhen

j = 0

andthesamplinginterval

∆

isxed,andtheir remaindertermissmaller: itis

1/n

andnot

ln

2

_(n)/n

.

The penalty fun tion depends on

β

0

and

θ

. Unfortunately, these two onstantsare di ult to estimate. However,the slope heuristi dened in Arlotand Massart [1℄ enablesus to hoose

automati allya onstant

λ

su hthatthepenalty

λD

2j+1

m

/(n∆)

isgood. Itis alsopossibletouse theresamplingpenaltiesofLerasle[18℄.

2.5 Risk of the adaptive estimator on

R

Letus denote

M

j,n

= {m, 2

m

_{≤ D}

j,n

}

with

D

2j+2

j,n

≤ n∆ ∧ n

andx

N = N

n

= (n ∧ n∆)

. For any

m ∈ M

j,n

,anestimator

g

ˆ

j,m,N

n

∈ S

m,N

n

of

g

j

is omputed. Thebestdimension

m

ˆ

j

is hosen su hthat

ˆ

m

j

= arg min

m∈M

j,n

[γ

j,n

(ˆ

g

j,m,N

n

) + pen

j

(m)]

where

pen

j

(m) = cψ

j

 2

(2j+1)m

n

∨

2

(2j+1)m

nθ∆



andtheresultingestimatorisdenoted by

˜

g

j

:= ˆ

g

j, ˆ

m

j

,N

n

.

Theorem2.4: Adaptive estimationon

R

. UnderAssumptionsM1-M3 andS2,

E



k˜g

j

− g

j

k

2

_L

2



≤ C

inf

m∈M

j,n



kg

j,m

− g

j

k

2

_L

2

+ pen

j

(m)



+

c

n



1 ∨

_∆

1



where

c

dependson

ψ

j

,

β

0

and

θ

.

3 Case of stationary diusion pro esses

Letus onsiderthesto hasti dierentialequation(SDE):

dX

t

= b(X

t

)dt + σ(X

t

)dW

t

,

X

0

= η,

(3.1) where

η

isarandomvariableand

(W

t

)

t≥0

aBrownianmotionindependentof

η

. Thedriftfun tion

b : R → R

isunknownandthediusion oe ient

σ : R → R

+∗

isknown. Thepro ess

(X

t

)

t≥0

is assumedtobestri tlystationary,ergodi and

β

-mixing. Obviously,we an onstru testimatorsof thesu essivederivativesofthestationarydensityusingthepreviousse tion. Butinthisse tion,

weusethe properties ofadiusion pro ess to ompute anewestimatorof therstderivativeof

thestationarydensity. Ifthesamplinginterval

∆

issmall,thisnewestimator onvergefasterthan thepreviousone.

3.1 Model and Assumptions

Thepro ess

(X

t

)

t≥0

isobservedatdis retetimes

t = 0, ∆, . . . , n∆

. AssumptionM4.

Thefun tions

b

and

σ

are globally Lips hitzand

σ ∈ C

1

.

AssumptionM4ensurestheexisten eanduniqueness ofasolutionof theSDE(3.1).

AssumptionM5.

Thediusion oe ient

σ

belongsto

C

1

,isboundedandpositive: thereexist onstants

σ

0

and

σ

1

su hthat:

∀x ∈ R, 0 < σ

1

≤ σ(x) ≤ σ

0

.

AssumptionM6.

Thereexist onstant

r > 0

and

1 ≤ α ≤ 2

su hthat

UnderAssumptionsM4-M6,thereexistsastationarydensity

f

fortheSDE(3.1),and

f (x) ∝ σ

−2

(x) exp



2

Z

x

0

b(s)σ

−2

(s)ds



.

(3.2)

Then

f

hasmomentsofanyordersand:

Z

|f

′

(x)|

2

dx < ∞,

∀m ∈ N,

Z

|x|

m

|f

′

(x)| dx < ∞

(3.3)

∀m ∈ N, kx

m

_{f (x)k}

∞

< ∞,

b

4

_{(x)f (x)}

∞

< ∞

and

Z

exp (|b(x)|) f(x)dx < ∞.

(3.4) AssumptionM7.

Thepro ess isstationary:

η ∼ f

.

A ordingto Pardouxand Veretennikov [21℄, Proposition 1 p.1063,under Assumptions

M5-M6,thepro ess

(X

t

)

isexponentially

β

-mixing: thereexist onstants

β

0

and

θ

su hthat

β

X

(t) ≤

β

0

e

−θt

. Moreover,Gloter[11℄provethefollowingproperty: Proposition 3.1.

Let us set

F

t

= σ (η, W

s

, s ≤ t) .

Under Assumptions M4 andM7, for any

k ≥ 1

,there exists a onstant

c(k)

dependingon

b

and

σ

su hthat:

∀h, 0 < h ≤ 1, ∀t ≥ 0 E

sup

s∈[t,t+h]

|b(X

s

) − b(X

t

)|

k











F

_t

!

≤ c (k) h

k/2



1 + |X

t

|

k



.

Remark3.1.Toestimate

f

′

,itisenoughtohaveanestimatorof

2bf

andanestimatorof

f

. Indeed, a ordingtoequation(3.2),therstderivative

f

′

satises:

f

′

_(x)

f (x)

∝

2b(x)

σ

2

_(x)

− 2

σ

′

_(x)

σ(x)

.

Byassumption,the diusion oe ient

σ

is known. Besides, a ordingto AssumptionsM4and M5,

σ

′

and

σ

−1

arebounded. Aswehavealready onstru tedanestimatorof

f = g

0

in Se tion 2,itremainstoestimate

2bf

.

Inthisse tion,we onstru tanestimator

h

˜

of

h := 2bf

eitherona ompa tset

[a

0

, a

1

]

,oron

R

.

3.2 Sequen e of linear subspa es

Likein the previous se tion, estimators

ˆ

h

m

of

h

are omputed on some linear subspa es

S

m

or

S

m,N

, thenapenaltyfun tion

pen(m)

isintrodu edto hoosethebest possibleestimator

˜

h

. If

h

isestimatedona ompa tset

A = [a

0

, a

1

]

,thefollowingassumptionisneeded:

AssumptionS3: Estimationon a ompa t set. 1. Thesequen eoflinearsubspa es

S

m

is in reasing,

D

m

= dim(S

m

) < ∞

and

∀m

,

S

m

⊆ L

2

_(A)

.

2. There existsanorm onne tion: for any

m ∈ N

,any fun tion

t ∈ S

m

satises

ktk

2

∞

≤ φ

0

D

m

ktk

2

L

2

.

Parti ularly, ifwe note

Φ

m

(x) =

P

λ∈Λ

m

(ϕ

λ,m

(x))

2

where

(ϕ

λ,m

, λ ∈ Λ

m

)

is an orthonor-malbasis of

S

m

,then

Φ

2

_m

(x)

∞

≤ φ

0

D

m

.

3. There exists

r ≥ 1

su hthat,for anyfun tion

t

belonging to

B

α

2,∞

with

α ≤ r

,

kt − t

m

k

2

L

2

≤ D

_m

−2α

where

t

m

isthe orthogonal proje tionof

t

over

S

m

.

aregiven. Toestimate

h

on

R

,anin reasingsequen eoflinearsubspa es

S

m

=

Ve t

(ϕ

λ,m

λ ∈ Z)

(where

{ϕ

λ,m

}

λ∈Z

isan orthonormalbasisof

S

m

)is onsidered. As thedimensionof those sub-spa esisinnite,thetrun atedsubspa es

S

m,N

=

Ve t

(ϕ

λ,m

, λ ∈ Λ

m,N

)

areused.

AssumptionS4: Estimationon

R

. 1. The sequen eof linearsubspa es

(S

m

)

isin reasing. 2. Thedimension of the subspa e

S

m,N

is

2

m+1

_{N + 1}

. 3.

∃φ

0

,

∀m

,

∀t ∈ S

m

,

ktk

2

∞

≤ φ

0

2

m

ktk

2

L

2

.

Let us set

Φ

m

(x) =

P

λ∈Z

(ϕ

λ,m

(x))

2

, then

Φ

2

m

(x)

∞

≤ φ

0

2

m

where

φ

0

isa onstantindependent of

N

. 4. Foranyfun tion

t ∈ L

2

_{∩ L}

1

_(R)

su h that

R x

2

_t

2

_{(x)dx < +∞}

,

kt

m

− t

m,N

k

2

_L

2

≤ c

2

m

N

where

t

m

isthe orthogonal (

L

2

) proje tion of

t

over

S

m

and

t

m,N

itsproje tion over

S

m,N

. 5. There exists

r ≥ 1

su h that for any fun tion

t

belonging to the unit ball of a Besov spa e

B

_2,∞

α

with

α ≤ r

,

kt − t

m

k

2

_L

2

≤ c2

−2mα

.

Proposition 3.2.

Letus onsider afun tion

ϕ

generatinga

r

-regularmulti-resolutionanalysis of

L

2

with

r ≥ 0

. Let usset

S

m

=

Ve t

{ϕ

λ,m

, λ ∈ Z}

and

S

m,N

=

Ve t

{ϕ

λ,m

, λ ∈ Λ

m

}

where

ϕ

λ,m

(x) = 2

m/2

_{ϕ (2}

m

_{x − λ)}

and

Λ

m

= {λ ∈ Z, |λ| ≤ 2

m

_{N }}

. Then the subspa es

S

m,N

satisfyAssumptionS4.

Fun tions

ϕ(x) = sin(x)/x

also generate a multi-resolution of

L

2

_(R)

, but they are not even

0-regular. However,theysatisfyAssumptionS4ifSobolevspa estakethe pla eofBesovspa esin

Point5. Thedenition of Sobolevspa esof regularity

α

isre alledhere:

W

α

=



g,

Z

∞

−∞

|g

∗

_(x)|

2

x

2

_{+ 1}

α

dx < ∞



where

g

∗

istheFouriertransformof

g

.

3.3 Risk of the estimator with

m

xed

Forany

m ∈ M

n

, where

M

n

= {m, D

m

≤ D

n

}

, an estimator

ˆ

h

m

of

h = 2bf

is omputed. The maximaldimension

D

n

isspe iedlater. Thefollowing ontrastfun tionis onsidered:

Γ

n

(t) = ktk

2

L

2

−

4

n∆

n

X

k=1

X

(k+1)∆

− X

k∆

t (X

k∆

) .

As

∆

−1

_X

(k+1)∆

− X

k∆

= I

k∆

+ Z

k∆

+ b(X

k∆

)

with

I

k∆

=

1

∆

Z

(k+1)∆

k∆

(b(X

s

) − b(X

k∆

)) ds

and

Z

k∆

=

1

∆

Z

(k+1)∆

k∆

σ(X

s

)dW

s

,

(3.5) wehavethat

E

(Γ

n

(t)) = ktk

2

L

2

−4 hbf, ti−4E (I

∆

t(X

∆

)) .

A ordingtoLemma6.4,

|E (I

k∆

t(X

k∆

))| ≤

c∆

1/2

. Moreover,

h = 2bf

,so

E

_(Γ

_n

_{(t)) = ktk}

2

L

2

− 2 hh, ti + O



∆

1/2



_.

Thisinequalityjustiesthe hoi eof the ontrastfun tion ifthesampling interval

∆

issmall. If AssumptionS3issatised,we onsidertheestimator

ˆ

h

m

= arg min

t∈S

m

ˆ

h

m,N

= arg min

t∈S

m,N

Γ

n

(t).

Theorem3.1: Estimationon a ompa t set.

UnderAssumptionsM4-M7 andS3,

E



ˆ

h

m

− h

2

L

2



≤ kh

m

− hk

2

L

2

+ c∆ +



σ

2

0

kfk

∞

+

2β

0

φ

0

θ

 D

m

n∆

where

h

m

is the orthogonal proje tion of

h

over

S

m

and

c

a onstant depending on

b

and

σ

. We remindthat the

β

-mixing oe ientof the pro ess

(X

t

)

issu hthat

β

X

(t) ≤ β

0

e

−θt

.

Theorem3.2: Estimationon

R

. UnderAssumptionsM4-M7 andS4

E



ˆ

h

m,N

− h

2

L

2



≤ kh

m,N

− hk

2

_L

2

+ c

2

m

N

+ c∆ +



kfk

∞

+

2β

0

φ

0

θ

 2

m

n∆

.

where

h

m,N

isthe orthogonal proje tion of

h

onthe spa e

S

m,N

. If

N = N

n

= n∆

,then

E



ˆ

h

m,N

n

− h

2

L

2



≤ kh

m

− hk

2

L

2

+ c∆ +



kfk

∞

+

2β

0

φ

0

θ

 2

m

n∆

where

h

m

isthe orthogonal proje tion of

h

over

S

m

.

3.4 Optimisation of the hoi e of

m

UnderAssumptionS3,if

h

1

A

belongstotheunit ballofaBesovspa e

B

α

2,∞

,then

kh − h

m

k

2

L

2

≤

D

−2α

m

. Tominimisethebias-varian e ompromise,onehaveto hoose

D

m

∼ (n∆)

1/(1+2α)

andinthat asetheestimatorrisksatises:

E



ˆ

h

m

− h

2

L

2



≤ C (n∆)

−2α/(1+2α)

+ c∆.

UnderAssumptionS4,if

h

belongsto

B

α

2,∞

, then

kh − h

m

k

2

L

2

≤ 2

−2mα

and

E



ˆ

h

m,n∆

− h

2

L

2



≤ C (n∆)

−2α/(1+2α)

+ c∆.

Remark 3.2.Dalalyan and Kutoyants [9℄ estimate the rst derivative of the stationary density

observed at ontinuous time (they observe

X

t

for

t ∈ [0, T ]

). In that framework, the diusion oe ient

σ

2

isknown. Theminimaxrateof onvergen eoftheestimatoris

T

−α/(1+2α)

. Itisthe

ratethat weobtainwhen

∆

tendsto 0. Letusset

∆ ∼ n

−β

. Weobtainthefollowing onvergen etable:

β

prin ipaltermofthebound rateof onvergen eoftheestimator

0 < β ≤

2α

4α+1

∆

n

−β

2α

4α+1

≤ β < 1

(n∆)

−2α/(1+2α)

n

−2α(1−β)/(4α+1)

Thoseratesof onvergen earethesameasfortheestimatorofthedrift. If

β ≥ 1/2

,the domi-natingtermintheriskboundisalways

(n∆)

−2α/(1+2α)

. Therateof onvergen eisalwayssmaller

than

n

−1/2

. If

(n, ∆)

is xed and if

∆ ≤ n

−2α/(4α+3)

, then these ond estimator

h

ˆ

m

onverges fasterthantherstone

ˆ

g

1,m

. However,ifthesamplinginterval

∆

islargerthan

n

−2α/(4α+3)

,itis

For any

m ∈ M

n,A

= {m, D

m

≤ D

n

}

where the maximal dimension

D

n

is spe ied later, an estimator

ˆ

h

m

∈ S

m

of

h

is omputed. Letusset

pen(m) ≥ κ

D

m

n∆



1 +

8β

0

θ



and

m =

ˆ

inf

m∈M

n,A

n

γ

n

ˆh

m



+ pen(m)

o

.

Theresultingestimatorisdenoted by

˜

h := ˆ

h

m

ˆ

. Letus onsidertheasymptoti framework: AssumptionS5.

n∆

ln

2

(n)

→ ∞

and

D

2

n

≤

n∆

ln

2

(n)

.

Theorem3.3: Adaptive estimationon a ompa t set.

There exists a onstant

κ

depending only on the hosen sequen e of linear subspa es

(S

m

)

su h that,underAssumptionsM4-M7 ,S3andS5,

E



˜

h − h

2

L

2



≤ C

inf

m∈M

n,A

n

kh

m

− hk

2

L

2

+ pen(m)

o

+ c∆ +

c

′

n∆

where

C

isanumeri al onstant,

c

′

depends on

φ

0

and

kfk

∞

and

c

depends on

b

.

Remark 3.3.The estimator is only onsistent if

∆ → 0

. Moreover, the adaptive estimator

˜

h

automati allyrealisesthebias-varian e ompromise.

3.6 Risk of the adaptive estimator on

R

An estimator

ˆ

h

m,n∆

∈ S

m,n∆

is omputed for any

m ∈ M

n,R

= {m, 2

m

≤ D

n

}

. Thefollowing penaltyfun tion isintrodu ed:

pen(m) ≥ κ

2

m

n∆



1 +

2β

0

θ



andweset

m = inf

ˆ

m∈M

n

n

γ

n

ˆh

m,n∆



+ pen(m)

o

Letus denoteby

˜h

n∆

theresultingestimator. Theorem3.4: Adaptive estimationon

R

.

There existsa onstant

κ

depending only on the sequen e of linear subspa es

(S

m

)

su h that, if AssumptionsM4-M7,S4andS5aresatised:

E



˜

h

n∆

− h

2

L

2



≤ C

inf

m∈M

j,n,R

n

kh

m

− hk

2

_L

2

+ pen(m)

o

+ c∆ +

c

′

n∆

.

4 Drift estimation by quotient

Ifthepro ess

(X

t

)

t≥0

isthesolutionofthesto hasti dierentialequation(SDE)

dX

t

= b(X

t

)dt + dW

t

andsatisesAssumptionsM4-M7,then

b = f

′

_/2f.

An estimator of the drift by quotient an therefore be onstru ted. For high-frequen y data,

Comteetal.[6℄buildanadaptivedriftestimatorthankstoapenalizedleast-squaremethod. Their

estimator onvergeswiththeminimaxrate

(n∆)

−2α/(2α+1)

if

b

belongstotheBesovspa e

B

α

2,∞

. Onthe ontrary, there exist fewresultson thedriftestimation where thesampling interval

∆

is xed. Gobet etal.[12℄ buildadriftestimatorforlow-frequen ydata, however,theirestimatoris

noteasyto implement. Inthisse tion, adriftestimatorbyquotientis onstru tedanditsriskis

Weestimate

f

and

f

′

on

R

inordertoavoid onvergen eproblemsontheboundariesofthe om-pa t. Letus onsider twosequen esof linearsubspa es

(S

0,m

, m ∈ M

0,n

)

and

(S

1,m

, m ∈ M

1,n

)

satisfyingAssumptionS2for

k = 1

and su hthat

M

_0,n

₌

n

_m

₀

_{, log(n) ≤ 2}

m

0

≤ η

√

n∆/ log(n∆)

o

and

M

1,n

=

n

m

1

, 2

m

1

≤ (n∆)

1/5

o

wherethe onstant

η

doesnotdependon

b

neither

σ

.

As in Se tion 2, adaptive estimators

f := ˜

˜

g

0,n∆

and

g := ˜

˜

g

1,n∆

of

f = g

0

and

f

′

_{= g}

1

are omputed. As

b

belongs to

B

α

2,∞

,

f

and

f

′

alsobelong to

B

α

2,∞

and the best bias-varian e ompromisefor

g

ˆ

0,m

is obtained for

2

m

0

_{∼ (n∆)}

1/(1+2α)

, and for

ˆ

g

1,m

it is obtained for

2

m

1

_∼

(n∆)

1/(3+2α)

. If

α > 1

, therestri tionson

M

0,n

and

M

1,n

donotmodifytherateof onvergen e ofoursestimators. Letus onsidertheestimator

˜b =

˜

g

2 ˜

f

if

g ≤ 2n∆ ˜

˜

f

and

˜b = 0

otherwise

.

Theorem4.1. If

b ∈ B

α

2,∞

with

α > 1

,then

E



˜b − b

2

L

2



≤ c



E



˜

f − f

2

L

2



+ E



k˜g − gk

2

L

2



+

1

n∆



wherethe onstant

c

does notdepend on

n

nor on

∆

. Then, byTheorem2.4,

E



˜b − b

2

L

2



≤ c(n∆)

−2α/(2α+3)

So

˜b

onvergestowards

b

withtheminimaxratedenedbyGobetetal.[12℄.

5 Simulations

5.1 Models

Ornstein-Uhlenbe k: Letus onsidertheSDE

dX

t

= −bX

t

+ dW

t

with

b > 0

. Thestationary densityisaGaussiandistribution

N



0, (2b)

−1



anditsderivativeis

f

′

(x) = −

2b

3/2

√

π

xe

−bx

2

.

Hyperboli tangent: We onsiderapro ess

(X

t

)

satisfyingtheSDE

dX

t

= −a tanh(aX

t

)dt + dW

t

.

ThestationarydensityrelatedtothisSDEis

f (x) =

a

2 cosh

2

(ax)

and

f

′

(x) = −

a

2

_tanh(ax)

cosh

2

(ax)

.

Square root: Letus onsiderthediusionwithparameters

b(x) = −

√

ax

1 + x

2

and

σ = 1.

Thestationarydensityis

f (x) = c exp



−2a

p

1 + x

2



and

f

dX

t

= −

2aX

t

1 + X

2

t

dt + dW

t

.

Thepro ess

(X

t

)

t≥0

doesnotsatisfyAssumptionM6 neitherthesu ient onditionstobe expo-nentially

β

-mixing. If

a > 1/2

,itadmitsthestationary density

f (x) = c

a

1 + x

2

−2a

and

f

′

(x) = −

4c

a

ax

(1 + x

2

₎

1+2a

.

Sinefun tion: Letus onsiderthediusion withparameters:

b(x) = sin(ax) −

√

x

1 + x

2

and

σ = 1.

Itsstationarydensity

f

satises:

f (x) = c

a

exp



−2a

−1

cos(ax) − 2

p

1 + x

2



and

f

′

_{(x) = 2c}

a

b(x)f (x)

5.2 Estimation of the rst derivative

f

′

Here,weestimatetherstderivative

f

′

ofthestationarydensityona ompa tsetandwe ompare

the two estimators

˜

g

1

and

˜

h

dened in Se tions 2 and 3. The subspa es

S

m

are generated by trigonometri polynomials: those fun tions are orthonormal, very regular and enable very fast

omputations: to ompute

ˆ

g

1,m

(resp

ˆ

h

m

)when

g

ˆ

1,m−1

(resp

ˆ

h

m−1

)isknown,itisonlyne essary to omputeoneortwo oe ients.

Figures1-5 showthedieren es betweenthetwoestimators:

g

˜

1

onvergeswhateverthe sam-plinginterval,and

˜

h

onvergesonlyif

∆

is small. Inthat ase,

˜

h

is better than

g

˜

1

: thevarian e termisgreaterfor

g

ˆ

1,m

(isproportionalto

D

3

m

/(n∆

))thanfor

ˆ

h

m

(ispproportionalto

D

m

/n∆

). InTables1-3,forea hvalueof

n

and

∆

,50exa tsimulationsofadiusionpro essarerealized using the retrospe tive exa t algorithm of Beskos et al. [3℄ (ex ept for the Ornstein-Uhlenbe k

pro ess whi h is simulatedusing Gaussian variables). Forea h path, we ompute the empiri al

risksoftheestimators

g

˜

1

and

˜

h

:

k˜g

1

− g

1

k

2

E

:=

1

M

M

X

k=1

(˜

g

1

(x

k

) − g

1

(x

k

))

2

and

h − h

˜

2

E

:=

1

M

M

X

k=1

˜h(x

k

) − h(x

k

)



2

,

wherethepoints

x

k

areequidistributedover

A

. To he kthattheestimatorisadaptive,theora les

or

g

=

k˜g

1

− g

1

k

2

E

min

m∈M

n

kˆg

1,m

− g

1

k

2

E

and

or

h

=

˜

h − h

2

E

min

m∈M

n

ˆ

h

m

− h

2

E

are omputed. The mean time of simulation

t

sim

of a pro ess is measured, and for ea h kind of estimator, the means of the empiri al risk

ris

g

or

ris

h

, of the ora les

or

¯

g

or

or

¯

h

and of the omputationtimes

t

g

or

t

h

or omputed.

The omplexity of the retrospe tive exa t algorithm of Beskos et al. [3℄ is proportional to

ne

c∆

where

c

depends on the model. Table3 showsthat forModel 4,

t

sim

in reaseswhen

n

or

∆

in reases. For the hyperboli tangent, thetime of simulationonly depends on

n

be ause the onstant

c

is exa tlyequalto 0. TheOrnstein-Uhlenbe kpro ess isnotsimulated thanksto the retrospe tivealgorithm,soitstimeofsimulationdoesnotdependon

∆

. Tables1-3showthat the rstestimator

g

˜

1

isalwaysfaster to omputethanthe se ond one

˜

h

. This ismainly be ausewe havelessmodelstotest: fortherstestimator,themaximaldimension

D

n

isboundedby

(n∆)

1/4

whereasforthese ondestimator,

D

n

≤ (n∆)

1/2

.

When

∆ = 1, ˜

g

1

isbetterthan

˜

h

. Ifnot,the estimatorsaresimilar andbe omebetterwhen

n∆

in reases. FortheOrnstein-Uhlenbe kpro essandthehyperboli tangent,thepro ess

(X

t

)

t≥0

isexponentially

β

-mixingand

g

˜

1

isingeneralbetterthan

˜

h

. ForModel4,thepro ess

(X

t

)

isnot exponentially

β

-mixingandwhen

∆ < 1

,

h

˜

is(ingeneral)betterthan

˜

g

1

.

Figure1: Ornstein-Uhlenbe k: estimationof

f

′

n = 10

4

,

∆ = 1

n = 10

5

,

∆ = 10

−2

−3

−2

−1

0

1

2

3

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

−3

−2

−1

0

1

2

3

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Figure2: Hyperboli tangent: estimationof

f

′

n = 10

4

,

∆ = 1

n = 10

5

,

∆ = 10

−2

−5

−4

−3

−2

−1

0

1

2

3

4

5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

−4

−3

−2

−1

0

1

2

3

4

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Figure 3: Squareroot: estimationof

f

′

n = 10

4

,

∆ = 1

n = 10

4

,

∆ = 10

−1

−4

−3

−2

−1

0

1

2

3

4

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

−4

−3

−2

−1

0

1

2

3

4

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

- : truederivative

· · ·

: estimator

˜

g

1

(dierentiating anestimatorof

f

) -. : estimator

˜

h

(usingto

f

′

_{= 2bf}

Figure4: Model4: estimationof

f

′

n = 10

4

,

∆ = 1

n = 10

4

,

∆ = 10

−1

−4

−3

−2

−1

0

1

2

3

4

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

−4

−3

−2

−1

0

1

2

3

4

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

Figure5: Sinefun tion: estimationof

f

′

n = 10

4

,

∆ = 1

n = 10

5

,

∆ = 10

−2

−8

−6

−4

−2

0

2

4

6

8

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

−8

−6

−4

−2

0

2

4

6

8

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

- : truederivative

· · ·

: estimator

˜

g

1

(dierentiating anestimatorof

f

) -. : estimator

˜

h

(usingto

f

′

_{= 2bf}

Table1: Estimationof

f

′

forOrnstein-Uhlenbe k

rstestimator se ond estimator

n

∆

t

sim

ris

g

or

¯

g

t

g

ris

h

or

¯

h

t

h

10

4

1 0.10 0.00025 2.5 0.33 0.0090 1.0 0.73

10

4

₁₀

−1

0.10 0.0010 1.8 0.17 0.00091 1.2 0.68

10

4

₁₀

−2

0.099 0.0060 2.6 0.097 0.0067 2.3 0.66

10

3

1 0.0027 0.0023 4.2 0.034 0.0097 1.0 0.12

10

3

₁₀

−1

0.0025 0.0058 3.0 0.020 0.0077 2.3 0.12

10

3

10

−2

0.0026 0.037 3.0 0.0070 0.078 4.0 0.035

10

2

1 0.00022 0.0080 2.0 0.013 0.019 1.5 0.062

10

2

₁₀

−1

0.00021 0.035 2.4 0.0046 0.078 5.5 0.019

10

2

₁₀

−2

0.00023 0.067 2.1 0.0048 0.11 1.4 0.0068

Table2: Hyperboli tangent: estimationof

f

′

rstestimator se ondestimator

n

∆

t

sim

ris

g

or

¯

g

t

g

ris

h

or

¯

h

t

h

10

4

1 6.2 0.0027 1.1 0.33 0.0087 1.03 0.71

10

4

10

−1

1.2 0.0018 3.7 0.17 0.0014 1.4 0.68

10

4

₁₀

−2

1.7 0.0065 2.8 0.10 0.0056 1.8 0.65

10

3

1 0.61 0.0040 1.5 0.034 0.0097 1.1 0.12

10

3

₁₀

−1

0.19 0.0067 2.8 0.020 0.0087 2.1 0.12

10

3

₁₀

−2

0.16 0.022 2.5 0.0068 0.036 2.6 0.03

10

2

1 0.066 0.011 1.7 0.014 0.021 1.80 0.063

10

2

₁₀

−1

0.020 0.023 2.3 0.0048 0.044 3.4 0.020

10

2

₁₀

−2

0.018 0.033 1.6 0.0054 0.078 1.2 0.0080

Table3: Model4: estimation of

f

′

rstestimator se ond estimator

n

∆

t

sim

ris

g

or

¯

g

t

g

ris

h

or

¯

h

t

h

10

4

1 6.6 0.00073 1.8 0.33 0.020 1.0 0.71

10

4

₁₀

−1

2.3 0.0032 4.2 0.17 0.0019 1.3 0.70

10

4

₁₀

−2

2.1 0.016 3.8 0.10 0.0090 1.7 0.68

10

3

1 0.67 0.0049 2.4 0.035 0.022 1.1 0.12

10

3

₁₀

−1

0.24 0.017 3.6 0.021 0.013 2.0 0.12

10

3

₁₀

−2

0.18 0.043 2.0 0.0071 0.094 3.5 0.035

10

2

1 0.071 0.048 8.1 0.014 0.041 1.6 0.065

10

2

₁₀

−1

0.022 0.046 1.91 0.0049 0.077 3.1 0.02

10

2

₁₀

−2

0.019 0.070 1.4 0.005 0.12 1.1 0.0069

ris

g

and

ris

h

: averageempiri alrisksrelatedfor

g

˜

1

and

h

˜

¯

or

g

and

or

¯

h

: average ora les (empiri al risks of

g

˜

1

(resp

˜

h

) over the empiri al risk of the best estimator

g

ˆ

1,m

(resp

ˆ

h

m

))

t

g

et

t

h

: averagetimeof omputationof

g

˜

1

and

˜

h

(timesin se onds)

Twodrift estimatorsare ompared: theestimatorbyquotientdened in Se tion 4,denoted here

by

˜b

quot

, and a penalized least-square estimator denoted by

˜b

pls

. The onstru tion of the last estimatorisdoneinComteetal.[6℄. Itonly onvergeswhenthesamplinginterval

∆

issmall,but inthat ase,itrea hestheminimaxrateof onvergen e:if

b

belongstoaBesovspa e

B

α

2,∞

,then theriskoftheestimator

˜b

pls

isboundedby

E



˜b

pls

− b

2

L

2



≤ C



(n∆)

−2α/(2α+1)

+ ∆



.

Figures6-10showthat,forlow-frequen ydata,thequotientestimator

˜b

quot

isbetterthan

˜b

pls

. Forvariousvaluesof

n

and

∆

,50exa tsimulationsof

(X

0

, . . . , X

n∆

)

are realizedandestimators

˜b

quot

and

˜b

pls

are omputed. Table4and5givetheaverageempiri alriskfortheseestimatorsand theaverage omputationtimes. Thelowestriskissetin bold.

Tables 4 and 5 underline that the rst estimator is always faster than the se ond one: to

ompute

˜b

pls

, wehaveto inverseamatrix

m × m

overea h spa e

S

m

. When

∆

is smalland the timeofobservation

n∆

islarge,thepenalizedleastsquare ontrastestimator onvergesbetterthan thequotientestimator. Of ourse,when

∆

isxed,

˜b

quot

onvergesfasterthan

˜b

pls

.

Table4: Ornstein-Uhlenbe k: estimationof

b

quotientestimator least-squareestimator

n

∆

ris

quot

t

quot

ris

pls

t

pls

10

4

1 0.0022 3.6 0.089 7.3

10

4

₁₀

−1

0.0086 1.2 0.0049 1.7

10

4

10

−2

0.069 0.4 0.031 0.7

10

3

1 0.011 0.2 0.090 0.7

10

3

₁₀

−1

0.061 0.06 0.022 0.3

10

3

₁₀

−2

0.31 0.02 0.50 0.004

10

2

1 0.073 0.03 0.085 0.3

10

2

₁₀

−1

0.25 0.01 0.34 0.003

Table5: Hyperboli tangent: estimationof

b

quotientestimator least-squareestimator

n

∆

ris

quot

t

quot

ris

pls

t

pls

10

4

1 0.0023 3.6 0.086 7.2

10

4

₁₀

−1

0.019 1.2 0.017 1.8

10

4

₁₀

−2

0.078 0.4 0.052 0.7

10

3

1 0.036 0.2 0.18 0.7

10

3

₁₀

−1

0.12 0.06 0.065 0.3

10

3

₁₀

−2

0.17 0.02 0.61 0.004

10

2

1 0.24 0.03 0.10 0.3

10

2

₁₀

−1

0.20 0.01 0.53 0.003

ris

quot

and

ris

pls

: averageempiri alrisksfor

˜b

quot

and

˜b

pls

t

quot

and

t

pls

: average omputationtimesof

˜b

quot

and

˜b

pls

(timesin se onds)

6 Proofs

6.1 Important lemmas

Lemma6.1 : Varian e of

β

-mixing variables. Letus set

A =

1

n

n

X

k=1

g(X

k∆

) − E (g(X

k∆

)) .

Figure6: Ornstein-Uhlenbe k: estimationof

b

n = 10

4

,

∆ = 1

−1.5

−1

−0.5

0

0.5

1

1.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Figure7: Hyperboli tangent: estimationof

b

n = 10

4

,

∆ = 1

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Figure8: Squareroot: estimation of

b

n = 10

4

,

∆ = 1

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

-: truedrift

b

−−

: estimationof

b

byquotient:

˜b

quot

.. : estimation of

b

likein Comte etal.[6℄ :

˜b

pls

Figure9: Model 4: estimationof

b

n = 10

4

,

∆ = 10

−1

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

−1.5

−1

−0.5

0

0.5

1

1.5

Figure 10: Sine fun tion: estimationof

b

n = 10

4

,

∆ = 1

−4

−3

−2

−1

0

1

2

3

4

−1.5

−1

−0.5

0

0.5

1

1.5

-: truedrift

b

−−

: estimationof

b

byquotient:

˜b

quot

.. : estimation of

b

likein Comte etal.[6℄ :

˜b

pls