Nonparametric recursive density estimation for spatial data

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C. R.

Acad.

Sci.

Paris,

Ser. I

www.sciencedirect.com

Statistics

Nonparametric

recursive

density

estimation

for

spatial

data

Estimation nonparamétrique récursive de la densité pour données spatiales

Aboubacar

Amiri

a

,

Sophie

Dabo-Niang

a

,

b

,

Mohamed

Yahaya

a

,

c

a_{UniversitéLille3,LaboratoireLEMCNRS9221,France} b_{INRIA-MODALTeam,France}

c_{FST,UniversitédesComores,Comoros}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received26February2014

Acceptedafterrevision18October2015 Availableonline23December2015 PresentedbytheEditorialBoard

Thispaper dealswith non-parametricdensityestimationfor spatialdata. Westudythe asymptoticpropertiesofanewrecursiveversionoftheParzen–Rozenblattestimator.The meansquareerrorandanalmostsureconvergenceresultwithrateofsuchestimatorare derived.

©2015PublishedbyElsevierMassonSASonbehalfofAcadémiedessciences.Thisisan openaccessarticleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

r

é

s

u

m

é

Cepapiertraitedel’estimationdeladensitéspatialedanslecasrécursif.Nousétudionsles propiétésasymptotiquesd’unenouvelleversiondel’estimateurdeParzen–Rozenblatt.Nous établissonslesconvergencesenmoyennequadratiqueetpresquesûredecetestimateur ; desvitessesdeconvergencesontdonnées.

©2015PublishedbyElsevierMassonSASonbehalfofAcadémiedessciences.Thisisan openaccessarticleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

Weconsiderrecursivekerneldensityestimationforspatiallydependentdata.Spatialdensityestimationisaninteresting andcrucialprobleminstatisticalinferenceforanumberofapplications,wheretheconsideredvariableshavespatial depen-dence.Spatialdataaremodeledasfiniterealizationsofrandomfieldscollectedfromdifferentspatiallocationsontheearth andtheyare widelyusedina varietyoffields,includingsoilscience,geology,oceanography, econometrics,epidemiology, environmentalscience, forestry, andmanyothers,seeCressie [9],Chiles andDelfiner [7].Althoughpotential applications ofnonparametric spatialmodels areabundant, littletheoretical workhas beendevotedtononparametric spacemodeling comparedtotheparametriccase.

Thiswork concernsa nonparametricestimationof theprobability densityfunctionfromdependentspatial data,using a recursive kernel approach. For an overview on results and applications considering independent ordependent spatial data fordensityestimation by classical kernelmethod, we highlight theworks by Biau andCadre[3],Carbon et al.[5], Dabo-NiangandYao[10],Tran[19].Themethodandtheresultsobtainedheregeneralizethepreviousworksintherecursive framework.

E-mailaddresses:[email protected](A. Amiri),[email protected](S. Dabo-Niang),[email protected](M. Yahaya). http://dx.doi.org/10.1016/j.crma.2015.10.010

1631-073X/©2015PublishedbyElsevierMassonSASonbehalfofAcadémiedessciences.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Withtheadvancesinhardwaretechnologyandthesophisticationofmoderncomputinganddataacquisitiontechniques, voluminousspatialdataarecollectedinvariousappliedareas.Newchallengesarisesinceinmanyapplicationsthevolumeof thedataissolargethatitmaybeimpossibletostorethemondeskandtheirmodelingrequiresmanytime.Nonparametric estimatorssuchasthekerneldensityestimatorofParzen–Rozenblattcanbeusedtoestimatethedensityfunction.However, theysufferfromaseriouscomputationaldrawbackinbigdatacontext.Infact,ifthedensityisestimatedbyanon-recursive estimator,thislastmustbecompletelyrecalculatedwhennewdataitemsarereceived.

In situations wheredata itemsarrivesequentially anda largeamount ofdata canbe generated ata rapidrate, a re-cursive updating algorithms providegreat help for the difficulties arising from the large volume of data. Then they are muchpreferredovernon-recursiveones,sincerecursiveestimatorscanbeupdatedfromtheirimmediatepastandthenew observation. Therefore,theupdatecanbecomputedinstantlyanddoesnotrequireextensivestorageofdata.This arrange-ment isparticularto recursivemethodsandiscalledtheonlineorreal-timeupdatingproperty. Thisrecursive propertyis particularlyinterestingwhenthesampledataarecontinuallycapturedovertime.

Recursivekernelapproacheswere introducedandabundantlystudiedinthenon-spatialcase.Werefer,forinstance,to the pioneerworksby WolvertonandWagner [20],Deheuvels [12],Wegman andDavies [11], whohavepaid attentionto nonparametricrecursivedensityestimationfortimeprocesses.Therearesomerecentcontributionsonasymptoticproperties of recursive kernel densityfor dependent data. Amiri [1]generalized all the above-cited versions to a generalfamily of recursive estimators, usingdifferent valuesof a parameter

∈ [

0,1

]

associatedwitheach estimator. Forthe fixed values

=

1/2 and

=

1, the asymptotic normality under negative association was previously treated by Liang and Baek [2], while themean-squareconvergenceandtheasymptoticnormality werestudiedby Masry[14] under

α

-mixingcondition. Mezhoudetal.[15]haveextendedtheresultsobtainedbyAmiri[1]to

η

-dependenttimeprocess.

We extend the feature of time-varying sample recursive density estimate to the spatial case. Namely, we presenta recursive versionoftheclassical spatialkerneldensityestimatorandestablishsomeasymptoticresultsofsuchestimator. Asfarasweknow,theinvestigationofrecursivekernelmethodsforspatialdataremainanopenproblem.

The restof the paper isorganized asfollows. In section 2, we present the spatial recursive density estimator, while Section 3 gives general assumptions. In section 4, we establishasymptotic convergence, namely mean square error and almost-sure convergence with the rate of our estimator. Finally Section 5 is devoted to some indications to prove the theoreticalresultspresentedinthisnote.

2. Therecursivespatialkerneldensityestimate

Let

(

Xi

)

i∈NN

,

N

≥

1 be a measurable strictly stationary spatial process defined on a probability space

(,

A

,

P

)

and

valued in

R

d_,whered

≥

_{1.We assume that the X}

i’shave thesame distributionasa random variable X,withunknown

probability densityfunction p.Suppose that weobservethe process onaspatial set ofsites

I

n ofcardinal n,whichis a finite subsetofapotentially observableregion

D

⊂

R

N,where

R

N isendowedwiththeuniformmetric.Forconvenience, we treat the observationssites asan array that is

I

n

=

sj

,

j

=

1, . . . ,n

⊂

N

N_{. The estimator ofthe probability density} functionbasedon

(

Xi

,

i

∈

I

n

)

tobestudiedinthepresentworkisoftheform:

p_n

(

x0

)

:=

1 Sn,

i∈In 1 hd_i K

Xi

−

x0 hi

,

x0

∈

R

d

(

∈ [

0

,

1

]

),

(1) whereSn,

:=

i∈In

hd(_i 1−),hiisabandwidthcorrespondingtotheobservationXi.Byenumeratingthesites,onemayrewrite

p n

(

x0

)

as p_n

(

x0

)

:=

1 Sn, n

j=1 1 hdsj K

Xs_j

−

x0 hs_j

,

x0

∈

R

d

(

∈ [

₀

,

₁

]

),

₍₂₎ whereSn,

:=

n

j=1

hd(sj1−).Thisrepresentationoftheestimatorpresentsagreatinterestfromthecomputationalpointofview.

Infact,ifXsn+1 isanewobservationoftheprocessatasitesn+1 addedto

I

n,theestimatorcanbeupdatedrecursivelyby thefollowingformula:

p_n+1

(

x0

)

=

Sn, Sn+1, p_n

(

x0

)

+

Kn+1

Xsn+1

−

x0

with K_i

(.)

:=

1 hdsiSi, K

.

hsi

.

(3)

In(3),p_n+1

(

x0

)

isthedensityestimatorbasedonthedomain

I

n

∪ {

sn+1

}

.Thisrecursiveformulaisusefulwhenthenumber

ofthespatialsitesincreasessequentiallyonspace.

Tosimplifythepresentation,theparameter

isthroughoutsupposedtobe equalto0 andthecorrespondingestimator willbelaternoted pn.

Basically,twoasymptoticmethodsoccurinthespatialliterature:increasingdomainandinfillasymptotics,see[9],p. 480. The growthof thesample inincreasing domainasymptotics isa consequenceofan unbounded expansionof thesample

region

I

n,whereasunderinfillasymptoticsthesampleregionisfixedandthegrowthofthesamplesizeisduetosampling thatisdenseintheregion

D

.Inwhatfollowsweconsidertheincreasingdomainasymptoticsandforsimplicitythebivariate regularlattice(N

=

2),describedinthefollowingassumption.

H2.1.D isaregularlatticeand

I

n

= {

i

=

(

i1

,

i2

),

1

≤

ij

≤

nj

,

j

=

1,2

}

isrectangular.Inthiscase,n

=

n1

×

n2goestoinfinitymeans thatminn_{j j=1,2}goestoinfinity.Therefore,weusethelexicographicorderandrenumbertheobservationsasatriangulararrayinthe followingway.Theobservationsite

(

i

,

j

)

canbeindexedbyt

=

n2

(

i

−

1)

+

j inthetriangulararraysetting,see[17].Inthiscase,the newobservationsiteis

(

n1

+

1,1),whichisindexedbyn

+

1inthearraysetting.

Noticethattoobtainourasymptoticresultsintheirregularlatticecontext,onecanconsideradditionalassumptionson thenumberofspatialunitonaclosedballof

D

withthefollowing:

H2.2.Thepotentialobservableregion

D

isanirregularlatticeandinfinitecountable.Allelementsof

D

arelocatedatdistancesofat least

δ >

1fromeachother.

Inthiscase(asin[8]),weconsiderasequenceoffiniteclosed,convexregions

{

Ru

}

ofincreasingareaasu

→ +∞

andletthe samplesetconsistsoftheintersectionofoneoftheconvexregionsandD:

I

n

=

D

⊂

Ru.Wedonotspecifyhereanyorderandany otherassumptionontheconfigurationandgrowthofsamplesizeexcepttheassumptionofauniformlyincreasingareaof

{

Ru

}

inat leasttwonon-opposingdirectionstoincreasethesamplesizen (asu

→ +∞

),see[8].Thus,anewobservationcorrespondstoanynew sitesn+1of

D

notalreadyincludedin

I

n.

3. Generalassumptions

Inordertoestablishtheasymptoticresults,thefollowingassumptionswillbeconsidered.Letusconsiderthe represen-tationoftherecursivedensityestimatorgivenin(2),with

=

0.

H3.1.(a)hsn

↓

0andnh d+2 sn

→ ∞

asn

→ ∞

;(b)Forr

∈ ]−∞

,

2

+

d

]

,Bn,r

:=

1 n n

i=1

hsi hsn

r

→

Br

>

0.

H3.2.K isaboundedsymmetricdensitysuchthat

Rd u K

(

u

)d

u

=

0and

Rd

u

2K

(

u

)d

u

<

∞

. H3.3.Thedensityp isboundedandtwicecontinuouslydifferentiableon

R

d.

H3.4.Foranyi

,

j

∈ {

1,

. . . ,

n

}

suchthatsi

=

sj,therandomvector

(

Xsi

,

Xsj

)

hasadensity fsi,sjsuchthat:

sup

si=sj

gs

_i,sj

_∞

<

∞

,

where gsi,sj

=

fsi,sj

−

p

⊗

p

,

i

,

j

=

1

, . . . ,

n

.

H3.5.

(i) Thefield

(

Xsi

)

1≤i≤nis

α

-mixing:thereexistsafunction

φ

:

R

+

→

R

+with

φ (

t

)

0ast

→ ∞

,suchthatwheneverE

,

E

⊂

R

2 withfinitecardinalsCard

(

E

)

,Card

(

E

)

α

σ

(

E

) ,

σ

E

:=

sup

A∈σ(E),B∈σ(E)

|

P

(

A

∩

B

)

−

P

(

A

)

P

(

B

)

| ≤

ψ

Card

(

E

),

Card

(

E

)

φ

dist

(

E

,

E

)

,

where

σ

(

E

)

= {

Xi

,

i

∈

E

}

and

σ

E

=

Xi

,

i

∈

E

,dist

(

E

,

E

)

istheEuclideandistancebetweenE andEand

ψ(

·

)

isapositive symmetricfunctionnondecreasingineachvariable.

Thefunctions

φ

and

ψ

aresuchthat

φ (

i

)

≤

C i−θ_and

ψ(

_n

,

_m

)

≤

_Cmin(n

,

_m

)

_. (ii) For

>

0,letun

=

Ni=1

(log

ni

)(log log

ni

)

1+andassumethat

nlognθ4−N−2Nθ_h dθ θ−4N n u 2N 4N−θ n

→ ∞

,

where

θ >

max

4N

,

Nd

+

2 2

.

ThefirstconditionofH3.1isusualinnonparametricrecursiveestimation,whilethesecondoneiscrucialinourcalculus. This last is particularly used in the recursive kernel estimation, for instance by Masry [14], Wegman and Davies [11], Yamato[21],SamantaandMugisha[16]andIsogai[13].Ifhsn

=

Ann−

q _whereA

n

↓

A

>

0, 0

<

q

<

1,thentheassumption

H 3.1 is satisfied for all q such that rq

<

1 and Br

=

1

1

−

rq. Also the choice hsn

=

An

lnn n

q

,

0

<

q

<

1 2

+

d satisfies

assumption H 3.1. Assumptions H 3.2 and H 3.3 are classical in nonparametric estimation and easily verified by many kernelssuchasEpanechnikov,Gaussiankernels.

AbouttheassumptionH3.5(i),basically,inthespatialliteratureitissupposedthat

φ (

i

)

tendstozerowithpolynomial rate, or

φ (

i

)

≤

Cexp(

−

si

),

for someC

,

s

>

0,i.e.

φ (

i

)

tendstozerowithexponentialrate.Ourresultscanbeprovedunder theabovecondition.Finallyletusmentionthatthedefinitionofun inH3.5leadsto

n∈NN_n1_u

n

<

∞

(recallthatn

=

n). 4. Consistencyresults

Theorem 1belowestablishestheasymptoticmeansquareerroroftheestimatorpn viaitsvarianceandbias.

Theorem1.UndertheassumptionsH2.2 andH3.1–3.5,wehave: nhd_snVar

(

pn

(

x0

))

→

p

(

x0

)

B−_d1

Rd K2

(

u

)

du (4) and h−_s2 n

(

E

(

pn

(

x0

))

−

p

(

x0

))

→

B −1 d Bd+2

Rd K

(

u

)

2 d

j1,j2=1 uj1uj2

∂

2p

(

x0

)

∂

xj1

∂

xj2 du (5)

asn

→ ∞

,forallx0suchthatp

(

x0

)

>

0.Inparticular,ifhsn

=

n −1

4+d_{,wehavetheasymptoticbehavioraccordingtothemeansquare} error: n4+4d

E

(

_pn

(

_x0

)

−

_p

(

_x0

))

2

→

⎛

⎜

⎝

Rd K

(

u

)

d

j1,j2=1 uj1uj2

∂

2p

(

x0

)

∂

xj1

∂

xj2 du

⎞

⎟

⎠

2

+

4 4

+

dp

(

x0

)

Rd K2

(

u

)

du (6)

asn

→ ∞

forallx0suchthatp

(

x0

)

>

0.

Theorem 1isanextensionofthepreviousresultsobtainedinthetemporalcasebyAmiri[1]with

α

-mixingcondition. Also, asimilarresultwereprovedbyMezhoudetal.[15] fortemporal

η

-dependenceprocess.Now,westatethefollowing theoremthatestablishesthealmostconvergenceofpn.

Theorem2.UndertheassumptionsH2.1,H3.1–3.5,wehave:

|

pn

(

x0

)

−

p

(

x0

)

| =

O

logn nhdn

,

a

.

s

.

(7)

5. Briefoutlineofproofs 5.1. Theorem 1 Forx0

∈

R

d,let I1

=

1 B2 n,dn2h2snd n

i=1 Var

Zsi

andI2

=

1 B2 n,dn2h2snd

si=sj Cov

Zsi

,

Zsj

,

with Zsi

=

K

Xsi

−

x0 hsi

. Ontheonehand,underH3.2withthehelpoftheToeplitzLemma(see[18]),wereadilyhave

nhd_s n

|

I1

| →

p

(

x0

)

B −1 d

Rd K2

(

u

)

du asn

→ ∞

.

Ontheotherhand,forx0

∈

R

d,letussplit:I2

=

J1

+

J2,where J1

=

1 B2_n,dn2_h2d sn

si−sj≤cn Cov

(

Zs_i

,

Zs_j

)

and J2

=

1 B2_n,dn2_h2d sn

cn<si−sj Cov

(

Zs_i

,

Zs_j

),

cn beingasequencetendingtoinfinityasntendstoinfinity.Ifd

≥

3,then2d

>

d

+

2.Inthiscase,weconsider

ζ

∈

R

such that N

θ

−

1

< ζ

≤

2

d,andusingthedecreaseof

(

hsn

)

n,onemaywrite:

Cov

(

Zsi

,

Zsj

)

≤

h d sihdsj sup si=sj

g

_si,sj

∞

≤

h2_sd i

+

h 2d sj 2 ssupi=sj

g

_si,sj

∞

≤

hd(ζsi +1)h d(1−ζ ) s1

+

h d(ζ+1) sj h d(1−ζ ) s1 2 ssupi=sj

gsi,sj

_∞

.

If1

≤

d

≤

3,take

ζ

=

1 andmakeaspreviously.As

I

n isaregulardesign(seeforinstance[19]) Card

(

u

,

v

)

∈

I

_n2

:

u

−

v

≤

cn

=

O

nc_nN

.

Itfollowsthatnhd_sn

|

J1

| ≤

Bn,d(ζ+1) B2 n,d hd(s11−ζ )h dζ snc N n sup si=sj

gsi,sj

_∞

=

O

hdζsnc N n

.

Turningto J2,bytheBillingsleyinequality(see[4]),wehave

Cov(Zsi

,

Zsj

)

≤

4φ (

si

−

sj

)

K

2 ∞.Consequently,weget nhd_s n

|

J2

| ≤

4

K

2_∞ B2_n,dhdsn

(θ

−

1

)

c1_n−θ

=

O

c1−θ n hdsn

.

The choice ofcn

=

h− d(ζ+1) N+θ−1 sn

leads to nhd_sn

|

I2

| =

O

⎛

⎜

⎝

h d

(ζ (θ

−

1)

−

N

)

N

+

θ

−

1 sn

⎞

⎟

⎠

→

0 asn

→ ∞

, aswell as

ζ >

N

θ

−

1,andthe result(4)follows.Aboutthebiasterm,weapply theTaylorformulaandthedominatedconvergence,whichtogether with Toeplitz’lemma,leadstotheresult(5).

2

5.2. Theorem 2 LetussetGn

(

x0

)

=

pn

(

x0

)

−

E

pn

(

x0

)

=

n

i=1

iwith

i

=

1 Sn,0

Zsi

−

E

Zsi

.Next,leta

=

an bean integerandsplitthe randomvariables

i intoblocksasfollows:

U

(

1

,

n

,

j

,

x0

)

=

(2j

k+1)an ik=2jkan+1 k=1,...,N

i

;

U

(

2

,

n

,

j

,

x0

)

=

(2jk

+1)an ik=2jkan+1 k=1,...,N−1 2(jN

+1)an iN=(2jN+1)an+1

i

;

U

(

3

,

n

,

j

,

x0

)

=

(2j

k+1)an ik=2jkan+1 k=1,...,N−2 2(jN

−1+1)an iN−1=(2jN−1+1)an+1 (2jN

+1)an iN=2jNan+1

i

;

U

(

4

,

n

,

j

,

x0

)

=

(2j

k+1)an ik=2jkan+1 k=1,...,N−2 2(jN

−1+1)an iN−1=(2jN−1+1)an+1 2(jN

+1)an iN=(2jN+1)an+1

i

,

(8)

andsoon.FinallynotethatU

(2

N−1

,

n

,

j

,

x0

)

=

(2j

k+1)an ik=2jkan+1 k=1,...,N−1 2(jN

+1)an iN=2jNan+1

i

;

U

(2

N

,

n

,

j

,

x0

)

=

(2jk

+1)an ik=2jkan+1 k=1,...,N

i.

Foreachinteger1

≤

i

≤

2N_{,define E}

(

_n

,

_i

,

_x 0

)

=

r

k−1 jk=0 k=1,...,N

U

(

i

,

n

,

j

,

x0

),

thenonemaywriteGn

(

x0

)

=

2N+1

i=1 E

(

n

,

i

,

x0

).

Without

lossofgenerality,wewillshowtheresultfori

=

1. Because K is bounded, we get U

(1,

n

,

j

,

x0

)

≤

a

N nK∞

Bn,d 1 nhdn

. Consider the sequences

λ

n

=

nhdnlogn

1/2

!

, and ri

=

ni

/(2

an

),

i

=

1,

. . . ,

N.Wededucethat,fornlargeenough,

λ

n

|

U

(

1

,

n

,

j

,

x0

)

| =

O

⎛

⎜

⎝

"

logn nhdn

⎞

⎟

⎠

.

(9)

Enumerate the randomvariables U

(1,

n

,

j

,

x

)

in E

(

n

,

1,x

)

inan arbitrary mannerand refer tothem as U1

ˆ

,

. . . ,

U

ˆ

M, M

=

r1

...

rN.ByMarkov’sinequalityandLemma4.5of[6],thereexistr.v.’sU1

˜

,

. . . ,

U

˜

M independentofU1

ˆ

,

. . . ,

U

ˆ

M andsuchthat

ˆ

Ui

= ˆ

Ui

(

x0

)

hasthesamedistributionasU

˜

i

= ˜

Ui

(

x0

)

and:

P

U

ˆ

i

− ˜

Ui

>

≤

18

⎛

⎜

⎝

U

ˆ

i

∞

⎞

⎟

⎠

ψ (

n

,

aN_n

)φ (

an

).

(10)

Let

n

=

η

logn nhd

n

,where

η

isapositivenumber.Obviously

P

(

|

E

(

n

,

1

,

x0

)

|

>

n

)

≤

P

⎛

⎝

M j=1

Ui

˜

>

n 2

⎞

⎠

+

P

⎛

⎝

M j=1

Ui

ˆ

− ˜

Ui

>

n 2

⎞

⎠

.

(11)

UsingtheindependenceoftheU

˜

i’sandapplyingBernsteininequality,weget

P

⎛

⎝

M j=1

Ui

˜

>

n 2

⎞

⎠

≤

2 exp

−

λ

n

n 2

+

λ

_n2 4 M

i=1

E

U

ˆ

2_i

(

x0

)

≤

2 exp

−

η

logn 2

+

λ

2_n 4 M

i=1

E

U

ˆ

_i2

(

x0

)

.

Clearly,

λ

2 n 4 M

i=1

E

U

ˆ

_i2

(

x0

)

≤

λ

2n 4Var(pn

(

x0

))

≤

log(n

)

4 nh d nVar(pn

(

x0

)).

From(4),wededucethatfornlargeenough,nhd_nVar(pn

(

x0

))

≤

p

(

x0

)

B_d−1

Rd K2

(

u

)d

u.Itfollowsthat

λ

2_n 4 M

i=1

E

U

ˆ

2_i

(

x0

)

≤

log

(

n

)

p

(

x0

)

B−_d1 4

Rd K2

(

u

)

du

.

Hence

P

⎛

⎝

M j=1

U

˜

i

>

n 2

⎞

⎠

≤

n−δ_,where

δ >

_1,for

η

_{largeenough.Concerningthesecondtermintheright-hand-sideof(11),} from(10)onehas

U

ˆ

i

(

x0

)

∞

≤

aN n

K

∞ nhd n

.Weget,byMarkov’sinequality:

P

_M

i=1

U

ˆ

i

− ˜

Ui

>

n 2

≤

C M

a_nN

K

_∞ nhd n

_n−1

ψ (

n

,

a_nN

)φ (

an

),

then

P

_M

i=1

U

ˆ

i

− ˜

Ui

>

n 2

≤

C

1 hdn

n−1

ψ(

n

,

anN

)

(φ (

an

))

≤

C

1 hnd

n−1anNa−nθ. Letan

=

logn nhd n

−1/(2N) ,then

P

_M

i=1

U

ˆ

i

− ˜

Ui

>

n 2

≤

Cn 2N−θ 2N _logn θ−2N 2N _h −dθ 2N n

=

C

β

n. By hypothesis, we have nun

β

n

=

nlognθ4−N2−Nθ_h dθ θ−4N n u 2N 4N−θ n

4N−θ 2N

→

0, then Theorem 2 follows by the Borel–Cantelli lemma.

2

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