Estimation of the conditional tail index using a smoothed local Hill estimator

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Estimation of the conditional tail index using a

smoothed local Hill estimator

Laurent Gardes, Gilles Stupfler

To cite this version:

Laurent Gardes, Gilles Stupfler. Estimation of the conditional tail index using a smoothed local Hill

estimator. Extremes, Springer Verlag (Germany), 2014, 17 (1), pp.45-75. �hal-00739454v2�

lo al Hill estimator

Laurent Gardes

(1)

&GillesStuper

(2)

(1)

UniversitédeStrasbourg&CNRS,IRMA,UMR7501,7rueRenéDes artes,

67084StrasbourgCedex,Fran e

(2)

Universitéd'Aix-Marseille,CERGAM,15-19alléeClaudeForbin,

13628 Aix-en-Proven eCedex1,Fran e

Abstra t. Forheavy-taileddistributions,theso- alledtailindexisanimportantparameterthat

on-trolsthebehaviorofthetaildistributionandisthusofprimaryinteresttoestimateextremequantiles.

In this paper, the estimation of the tail index is onsidered in the presen e of a nite-dimensional

random ovariate. Uniformweak onsisten yandasymptoti normalityoftheproposedestimatorare

establishedandsomeillustrationsonsimulationsareprovided.

AMSSubje t Classi ations: 62G05,62G20,62G30,62G32.

Keywords: Heavy-tailed distribution, random ovariate,uniform onsisten y, pointwise

asymp-toti normality.

1 Introdu tion

Extreme value analysis has attra ted onsiderable attention in many elds of appli ation, su h as

hydrology, biology and nan e, for instan e. It fo uses on the study of random variables having

survivalfun tion

F

of theform

F (x) = x

−1/γ

_L(x)

, where

γ > 0

shall bereferredto asthetailindex

and

L

isaslowlyvarying fun tionat innity: namely,

L

satises, forall

λ > 0

,

L(λx)/L(x) → 1

as

x

goestoinnity. Clearly,

γ

drivesthetailbehaviorof

F

anditsknowledgeisne essaryif,forinstan e,

weareinterestedintheestimationofextremequantiles. Theestimationofthetailindexisthusoneof

the entraltopi sinextremevaluetheory: thisproblemhasbeenextensivelystudiedintheliterature.

Re ent overviewson univariate tailindex estimation an be found in the monographs ofBeirlant et

al. [2℄ and deHaanandFerreira[12℄. Themost popularsemi-parametri estimatorwasproposed by

Hill[15℄. Let

k

n

∈ {2, . . . , n}

and

Y

1,n

≤ . . . ≤ Y

n,n

betheordered statisti sasso iatedto thesample

H(k

n

) =

1

k

n

− 1

k

_X

n

−1

i=1

log

Y

n−i+1,n

Y

n−k

n

+1,n

.

(1)

In pra ti e, it is often useful to link the variable of interest

Y

to a ovariate

X

. In this situation,

the tail index depends on the observed value

x

of the ovariate

X

and shall be referred to, in the

following,asthe onditional tailindex. Itsestimationhasbeenaddressedinthere entextremevalue

literaturemostlyinthexeddesign ase,thatis,whenthe ovariatesarenonrandom. Smith[17℄and

DavisonandSmith[8℄ onsideredaregressionmodelwhileHallandTajvidi[13℄usedasemi-parametri

approa hto estimatethe onditional tailindex. Fully nonparametri methods havebeen onsidered

usingsplines(seeChavez-DemoulinandDavison[4℄),lo alpolynomials(seeDavisonandRamesh[7℄),

amovingwindowapproa h (seeGardes andGirard[9℄),oranearestneighborapproa h(see Gardes

andGirard[10℄),amongothers.

Despite the great interest in pra ti e, less attention has been paid to the random ovariate ase.

One an ite the works of Wang and Tsai [18℄, based on a maximum likelihood approa h, Daouia

et al. [5℄ who use a xed numberof non parametri onditional quantile estimators to estimate the

onditional tail index, later generalized in Daouia et al. [6℄ to a regression ontext with response

distributions belonging tothegeneralmax-domainofattra tionandGoegebeuretal.[11℄ whostudy

anonparametri regressionestimator.

TheaimofthispaperistoadaptHill'sestimatortothepresen eofarandom ovariate. Notethatthe

uniformweak onsisten yoftheproposedestimatorisestablishedwhile,inmostoftheaforementioned

studies,theauthorsonly onsideredpointwise onvergen e.

Therestofthepaperisorganisedasfollows. InSe tion2,wedeneour onditionaltailindexestimator.

Thetwomainresults(uniformweak onsisten yandasymptoti normality)arestatedinSe tion3and

asimulationstudy isprovidedinSe tion 4. TheproofsaregiveninSe tion5andintheAppendix.

2 Estimation of the onditional tail index

Let

(X

1

, Y

1

), . . . , (X

n

, Y

n

)

be

n

independent opies of a random pair

(X, Y ) ∈ S × R

where

S

is a

subsetof

R

d

,

d ≥ 1

,havingnonemptyinterior. Forall

x ∈ S

,weassumethat the onditionalsurvival

fun tion of

Y

given

X = x

is heavy-tailed with tail index

γ(x) > 0

. Equivalently (see Bingham et

al.[3℄),we onsiderthemodel:

(

M

)

X

hasaprobabilitydensityfun tion

f

on

R

d

withsupport

S

andforall

x ∈ S

,the onditional

survivalfun tion

F (.|x)

is ontinuous and de reasing. Moreover,the onditional quantile of

Y

given

X = x

issu hthat

where

F

−1

(.|x)

is theinversefun tion ofthe onditional survivalfun tion and

ℓ(.|x)

isaslowly

varyingfun tionat innity.

For

i = 1, . . . , n

,denotingby

X

∗

i

the ovariateasso iatedwiththeorderedstatisti

Y

n−i+1,n

,a

straight-forwardadaptationtotherandom ovariate aseofHill'sestimator(1) is:

H(x, k, h) =

1

M

k

(x, h) − 1

k−1

_X

i=1

log

Y

n−i+1,n

Y

n−k+1,n

I

{kX

∗

i

− xk ∨ kX

k

∗

− xk ≤ h},

(3)

if

M

k

(x, h) > 1

and

H(x, k, h) = 0

otherwise. In(3),

I

{.}

is theindi ator fun tion,

k.k

isanorm on

R

d

,

k = k

n

∈ {2, . . . , n}

,

h = h

n

isanonrandompositivesequen etendingto0atinnityand

M

i

(x, h) =

i

X

j=1

I

{kX

∗

j

− xk ≤ h}, i = 1, . . . , n,

isthenumberof ovariatesamong

X

∗

1

, . . . , X

i

∗

whi hlieintheball

B(x, h)

with enter

x

andradius

h

.

Clearly,the hoi eofthenumber

k

in (3)is ru ialsin e,formostvaluesof

k

,thestatisti

H(x, k, h)

is equalto 0. The behavior of

H(x, k, h)

as a fun tion of

k

is thus veryerrati . To over omethis

drawba k,weproposetoestimatethe onditionaltailindexbyanaverageon

k

ofthestatisti sdened

in (3):

bγ

a

(x, k

x

, h) =

1

k

x

− ⌊(1 − a)k

x

⌋ + 1

n

X

l=2

H(x, l, h)I{⌊(1 − a)k

x

⌋ ≤ M

l

(x, h) ≤ k

x

}, a ∈ [0, 1),

(4)

where

⌊z⌋ = max{j ∈ N|z ≥ j}

is theintegerpartof

z

and

k

x

isa positive integerbelonging to the

interval

[2/(1 − a), n]

. Clearlyforall

a ∈ [0, 1)

,if

M

n

(x, h) > ⌊(1 − a)k

x

⌋

,then

bγ

a

(x, k

x

, h) > 0

. The

parameter

a

ontrolsthenumberofstatisti s(3)takenintoa ountintheestimator(4). Forinstan e,

if

a = 0

andif

M

n

(x, h) > k

x

,onlyonestatisti havingtheform

H(x, l, h)

isusedto ompute(4).

We point out that in pra ti e,

k

x

is restri ted to the interval

[2/(1 − a), M

n

(x, h)]

, sin e

k

x

is the

number of statisti s

Y

n−i+1,n

, whose asso iated ovariates

X

∗

i

belong to the ball with enter

x

and

radius

h

,whi hareusedto ompute

bγ

a

(x, k

x

, h)

: seealsoSe tion4.

3 Main results

InthisSe tion,westatethetwomainresultsofthepaper: theuniformweak onsisten yandpointwise

asymptoti normality of

bγ

a

(x, k

x

, h)

on a ompa t subset

Ω

of the interior of

S

. To this aim, we

introdu e some assumptions. The following ondition spe ies the regularityof the onditional tail

index

γ

andoftheprobabilitydensityfun tion

f

ofthe ovariates.

(A1) Thefun tion

γ

ispositiveand ontinuouson

S

andtheprobabilitydensityfun tion

f

isapositive

Hölder ontinuousfun tionon

S

withexponent

β

f

∈ (0, 1]

.

Notethatthis onditionespe iallyimpliesthat,onthe ompa tset

Ω

,thefun tion

γ

andtheprobability

densityfun tion

f

areboundedfrom belowandabovebynitepositive onstants:

0 < γ := inf

x∈Ω

γ(x) ≤ sup

_x∈Ω

γ(x) =: γ < ∞

and

0 < f := inf

se ondvariable. Forall

u < v ∈ (0, 1)

,let

ω(u, v, x, h) = sup

α∈[u,v]

sup

kx

′

−xk≤h

|log q(α|x) − log q(α|x

′

_{)| .}

Weassumethat

(A2) Thereexists

δ > 0

su hthat

lim

n→∞

sup

_x∈Ω

ω(n

−(1+δ)

_{, 1 − n}

−(1+δ)

_{, x, h) = 0.}

Now,in ordertodealwiththeslowlyvarying fun tionin (2),weassumethat

(A3) Forall

x ∈ S

and

t ≥ 1

,

ℓ(t|x) = c(x) exp

Z

t

1

∆(u|x)

u

du



,

where

c(x) > 0

and

∆(.|x)

isanultimatelymonotoni fun tion onvergingto0atinnity.

Note that(A3)impliesinparti ular thatforall

x ∈ S

,

ℓ(.|x)

isanormalisedslowlyvaryingfun tion

(seeBinghametal.[3℄). Wealsointrodu ethenotation

∆

x

(z) :=

sup

u∈[z

−1

_,∞)

|∆(u|x)|.

We annowstatetheuniformweak onsisten yofourestimator.

Theorem1. Under model (

M

),assume that(A1), (A2) and(A3)hold. If

nh

d

_{/ log n → ∞}

,

inf

x∈Ω

min



k

x

log n

,

nh

d

k

x

log(nh

d

)



→ ∞, lim

t→0

sup

_x∈Ω

∆

x

(t) = 0,

andif thereexistsanite positive onstant

K

1

su hthat

sup

x∈Ω

sup

kx

′

_−xk≤h

|k

x

− k

x

′

| ≤ K

₁

,

then, if

a ∈ (0, 1)

,itholdsthat,as

n

goes toinnity,

sup

x∈Ω

|bγ

a

(x, k

x

, h) − γ(x)|

−→ 0.

P

It is straightforwardthat under ondition(A1), thenumber

M

n

(x, h)

of ovariateslyingin the ball

B(x, h)

issu h that

1

n

M

n

(x, h) =

1

n

n

X

i=1

I

{kX

i

− xk ≤ h} = P(kX − xk ≤ h)(1 + o

P

(1)) = Vh

d

f (x)(1 + o

P

(1)),

(5)

where

V

is the volume of the unit ball of

R

d

(see Lemma 3for auniform result). Thus, sin e

f

is

boundedfrombelowandabovebynitepositive onstants, ondition

nh

d

_{/ log n → ∞}

impliesthatfor

all

x ∈ Ω

,

M

n

(x, h)

goestoinnityin probability. Furthermore, ondition

inf

x∈Ω

min



k

x

log n

,

nh

d

k

x

log(nh

d

)



→ ∞

implies that for all

x ∈ Ω

,

⌊(1 − a)k

x

⌋ = (1 − a)k

x

(1 + o(1)) → ∞

and that, with arbitrary large

probability,wehave

k

x

< M

n

(x, h)

for

n

su ientlylarge. Hen e,for

n

largeenough,

bγ

a

(x, k

x

, h) > 0

forall

a ∈ [0, 1)

and

x ∈ Ω

.

Wenowwishto statethepointwiseasymptoti normalityof theestimatorat apoint

x ∈ S

. Tothis

aim,thefollowingassumptionisrequired:

(A4) Forall

x ∈ S

,thefun tion

|∆(.|x)|

isregularlyvarying withindex

ρ(x) < 0

i.e., forall

λ > 0

,

lim

t→∞

|∆(λt|x)|

|∆(t|x)|

= λ

ρ(x)

_.

Notethat onditions(A3)and(A4)entailthat

lim

t→∞

log ℓ(λt|x) − log ℓ(t|x)

∆(t|x)

=

λ

ρ(x)

_{− 1}

ρ(x)

,

(6)

whi histhestandardse ond-order ondition lassi allyusedtoprovetheasymptoti normalityoftail

index estimators. The asymptoti normality of our estimatoris obtained onditionally to the event

{M

n

(x, h) = m

x

}

. Note that forinstan e, under (A1)and from(5), atypi al sequen e

(m

x

)

inthis aseis

m

x

= Vf (x)nh

d

.

Theorem 2. Under model (

M

), assumethat (A1), (A3) and (A4) hold. If, as

n

goes toinnity,

k

x

→ ∞

,

k

x

/m

x

→ 0

,

k

1/2

x

ω(m

−1−δ

x

, 1 − m

−1−δ

x

, x, h) → 0

and

k

1/2

x

∆(m

x

/k

x

|x) → ξ(x) ∈ R

,thenfor

a ∈ [0, 1)

and onditionally tothe event

{M

n

(x, h) = m

x

}

onehas

k

x

1/2



bγ

a

(x, k

x

, h) − γ(x) −

∆(m

x

/k

x

|x)

1 − ρ(x)

AB(a, x)



d

−→ N (0, γ

2

_(x)AV(a)),

whereif

a ∈ (0, 1)

,

AB(a, x) =

1 − (1 − a)

1−ρ(x)

a(1 − ρ(x))

and

AV(a) =

2(a + (1 − a) log(1 − a))

a

2

,

andif

a = 0

,

AB(0, x) = 1

and

AV(0) = 1

.

Asexpe ted,theasymptoti biasisade reasingfun tionof

a

whiletheasymptoti varian eis

in reas-ing. For

a = 0

,wendba ktheasymptoti biasandvarian eofHill'sestimator.

4 Simulation study

Toassessthenite-sampleperforman eoftheproposed onditionaltail-indexestimator,some

simula-tionexperimentswere arried outusingthefollowingmodel: the onditional distribution fun tion of

Y

given

X = x

isgivenby

∀ y > 0, F (y|x) =



1 + y

−ρ/γ(x)



−1/ρ

,

where

X

isuniformly distributed on

S = [0, 1]

. The negativese ond-orderparameter

ρ

is hosento

onditionaltail-index

γ

is on erned,twosituationsare onsidered:

γ

1

(x)

=

1

3

+

1

8

sin(2πx)

and

γ

2

(x)

=

1

4



1 + exp(−60(x − 1/4)

2

_{)I{3x ∈ [0, 1]} + exp(−5/12)I{3x ∈ (1, 2]}}

+ (5 − 6x) (exp(−5/12)I{3x ∈ (2, 5/2]} − I{3x ∈ (5/2, 3]})



.

Notethat

γ

1

isinnitelydierentiableand

γ

2

is ontinuousbutnotdierentiableat

x ∈ {1/3, 2/3, 5/6}

.

Theaimofthissimulationstudyistoestimatethe onditionaltail-indexonagridofpoints

{x

1

, . . . , x

M

}

of

[0, 1]

. Asmall preliminary pra ti alinvestigation leadsto take

a = 3/7

whi h provides reasonable

performan esin alargerangeofsituations. This leavestwoparameterstobe hosen: thebandwidth

h

and thenumberofupperorderstatisti s

k

x

. Oursele tionpro edure forthese parametersgoesas

follows.

1) We hoose a grid

{h

1

, . . . , h

P

}

of possible values of

h

. In what follows, we let

bγ

i,j

(k) :=

bγ

3/7

(x

i

, k, h

j

)

. For ea h

i ∈ {1, . . . , M }

,

j ∈ {1, . . . , P }

and

k ∈ {q

i,j

+ 4, . . . , M

n

(x

i

, h

j

) − q

i,j

}

,

where

q

i,j

∈ N \ {0}

,weintrodu etheset

E

i,j,k

= {bγ

i,j

(l), l ∈ {k − q

i,j

, . . . , k + q

i,j

}}

. Fortwo

xedindi es

i

and

j

, ouraimisto sele tthenumberofupperorderstatisti s

k

i,j

inaregionof

stabilityfor

bγ

i,j

. Todothat, we omputethevarian eof theset

E

i,j,k

foreverypossiblevalue

of

k

. Wethenre ordthenumber

K

i,j

forwhi h thisvarian eisminimal. Morepre isely,

K

i,j

= arg min

k

1

2q

i,j

+ 1

k+q

_X

i,j

l=k−q

i,j



bγ

i,j

(l) − bγ

i,j

(k)



2

with

bγ

i,j

(k) =

1

2q

i,j

+ 1

k+q

_X

i,j

l=k−q

i,j

bγ

i,j

(l).

Hen e,foragivenpoint

x

i

andagivenbandwidth

h

j

,thesele tednumberofupperorderstatisti s

k

i,j

is pi kedin theset

{K

i,j

− q

i,j

, . . . , K

i,j

+ q

i,j

}

. We propose to re ord the value

k

i,j

su h

that

bγ

i,j

(k

i,j

)

isthemedianoftheset

E

i,j,K

i,j

. Forthesakeofsimpli ity,theestimate

bγ

i,j

(k

i,j

)

willbedenoted by

eγ

i,j

.

2) We now want to sele t a bandwidth that does not depend on

x

and whi h is su h that the

estimation arriedout for bandwidthsin its neighborhood doesnot show alargevarian e. To

a hieve that, we let

q

′

be a positive integer su h that

2q

′

_{+ 1 < P}

and we ompute for ea h

j ∈ {q

′

_{+ 1, . . . , P − q}

′

_}

thestability riterion

σ(j) =

1

M

M

X

i=1

σ

i

(j),

where,for

i ∈ {1, . . . , M }

,

σ

i

(j) =





1

2q

′

_{+ 1}

j+q

′

X

l=j−q

′

(eγ

i,l

− eγ

i,.

(j))

2





1/2

with

eγ

i,.

(j) =

1

2q

′

_{+ 1}

j+q

′

X

l=j−q

′

eγ

i,l

.

Wenextre ordtheinteger

J

su hthat

σ(J)

istherstlo alminimumoftheappli ation

j 7→ σ(j)

whi h is lessthan the average of the

σ(j)

, see Figure 1. In other words,

J = q

′

_{+ 1}

if

σ(.)

is in reasing,

J = P − q

′

if

σ(.)

isde reasingand

J = min







j

su hthat

σ(j) ≤ σ(j − 1) ∧ σ(j + 1)

and

σ(j) ≤

1

P − 2q

′

P −q

′

X

l=q

′

₊₁

σ(l)







(7)

otherwise, where for onvenien e we extend

σ

by setting

σ(q

′

_{) := σ(q}

′

_{+ 1)}

and

σ(P − q

′

_{) :=}

σ(P − q

′

_{+ 1)}

. Thesele tedbandwidthisthen

h

∗

_{= h}

J

.

Tosummarize,thebandwidthandthenumberofupperorderstatisti saresele tedinordertosatisfy

astability riterion. Thesele tedbandwidthis independentof

x

andis givenby

h

∗

_{= h}

J

where

J

is

dened in(7). Thesele tednumberofupperorderstatisti sisgiven,for

x = x

i

,by

k

∗

x

i

= k

i,J

.

This estimation pro edure is arried out on

N = 100

independent samples of size

n = 1000

. The

onditional tail-indexis estimatedon agridof

M = 35

evenlyspa ed points in

[0, 1]

. Regardingthe

sele tionpro edure,

P = 100

valuesof

h

rangingfrom 0.025to0.25aretested. Theparameter

q

i,j

is

hosensothat

2q

i,j

+ 1

isapproximatelyequalto

5%

of

M

n

(x

i

, h

j

)

and

q

′

isset to3.

Tohaveanideaofhowourestimatorbehaves omparedtootherestimatorsinthe onditionaltail-index

estimationliterature,itis omparedto:

•

Theestimator

eγ

D

(x) := bγ

H

n

(x)

ofDaouiaetal.[5℄:

eγ

D

(x) =

P

9

j=1

[log b

q

n

(α

n

/j|x) − log b

q

n

(α

n

|x)]

P

9

j=1

log j

where

q

b

n

(α|x) = inf{t ∈ R, b

F

n

(t|x) ≤ α}

isthegeneralizedinverseofthekernelestimatorofthe

onditionalsurvivalfun tion

b

F

n

(y|x) =

n

X

i=1

K

h

(x − X

i

)I{Y

i

> y}

n

X

i=1

K

h

(x − X

i

)

.

Here

K

h

(x) = h

−1

_K(x/h)

where

K(x) =

15

16

(1 − x

2

₎

2

_I

_{[−1, 1](x)}

is thebi-quadrati kernel fun tion,

h := h

n

is apositive sequen etending to 0and

α

n

= 0.3

.

Thisestimatoris omputedusingthedata-drivenpro eduredes ribedin Daouiaetal.[5℄.

•

The estimator

eγ

G

(x) := bγ

(2)

n

(x, 0, K, K)

of Goegebeur et al. [11℄:

eγ

G

(x) := T

(1,1)

with

∀ s ≥ 1, ∀ t ≥ 0, T

(s,t)

n

(x) =

n

X

i=1

K

h

s

(x − X

i

)(log Y

i

− log ω

n,x

)

t

+

I

{Y

i

> ω

n,x

}

n

X

i=1

K

h

s

(x − X

i

)I{Y

i

> ω

n,x

}

.

Here

K

h

(u) = h

−1

_K(u/h)

where

K

is on e again thebi-quadrati kernel fun tion,

h := h

n

is

apositivesequen etending to 0and forall

x

,

(ω

n,x

)

is apositivesequen e tendingto innity.

Notethatthisestimatorisakernelversionofthe ase

a = 0

ofourestimator;to ompute

eγ

,we

shallusethedata-drivenmethoddes ribedin [11℄.

•

Thebias- orre tedversion

eγ

G,BC

(x) := bγ

(2)

n

(x, α

(2)

_BC

(b

ρ

n

(x; K, K, 0.5)))

of

eγ

G

(x)

,alsopresentedin

Goegebeuretal.[11℄:

eγ

G,BC

(x) =

bγ

(2)

n

(x, 0, K, K)

b

ρ

n

(x; K, K, 0.5)

+



1 −

1

b

ρ

n

(x; K, K, 0.5)



bγ

n

(2)

(x, 1, K, K)

where

bγ

n

(2)

(x, 1, K, K) =

T

n

(1,2)

(x)

2T

n

(1,1)

(x)

and

b

ρ

n

(x; K, K, 0.5) =

3(R

n

(x; K, K, 0.5) − 1)

R

n

(x; K, K, 0.5) − 3

provided

1 ≤ R

n

(x; K, K, 0.5) < 3

with

R

n

(x; K, K, 0.5) =



T

n

(1,1)

(x)

T

n

(1,0)

(x)



τ

−



T

n

(1,2)

(x)

2T

n

(1,0)

(x)



τ /2



T

n

(1,2)

(x)

2T

n

(1,0)

(x)



τ /2

−



T

n

(1,3)

(x)

6T

n

(1,0)

(x)



τ /3

.

Forthis estimator,thedata-drivenmethoddes ribedin [11℄isalsoused.

Foreveryestimator,we omputetheempiri alMSEs,averagedoverthe

M = 35

evenlyspa edpointsin

[0, 1]

. Numeri alresultsaregiveninTable1. This hartshowsthatourestimatoryieldsperforman es

whi haresimilarto theestimator

eγ

D

ofDaouiaetal.[5℄. Besides,itoutperformstheestimator

eγ

G

of

Goegebeuret al.[11℄in termsofMSEsby a2:1ratioin every ase,whilebeingoutperformedbythe

bias- orre tedversion

eγ

G,BC

ofthis estimator. This wasexpe ted, sin ethebias- orre tedestimator

eγ

G,BC

wasshowntodisplayfarbetterperforman esthanthesimpleestimator

eγ

G

andthatourmethod

wasnotoriginallytargetedat orre tinganyspe i biasthattheHillstatisti s

H(x, l, h)

usedforits

omputationmaypossess.

Wedisplay someresultsin Figures 24: theestimations orrespondingto the median,

10%

and

90%

quantilesof theMSE are represented. Besides, we representin Figure 5boxplotsof thebandwidths

and in Figure 6 boxplots of the ratios

k

∗

x

/M

n

(x, h

∗

)

at

x = 1/2

used to omputeour estimator. It

anbeseen that theestimator

bγ

a

generallyuses asmall bandwidth, whi h anbeinterpreted asan

Forthesakeofsimpli ity,weintrodu ethenotation

k

x,a

:= ⌊(1 − a)k

x

⌋

.

5.1 Proof of the uniform weak onsisten y

Weshallprovethat forall

ε > 0

,theprobability

p

n

:= P



sup

x∈Ω

|bγ

a

(x, k

x

, h) − γ(x)| > ε



,

onverges to 0 as

n

goes to innity. The proof is based on [14, Lemma 1℄: the basi idea is that

insteadofshowingtheuniform onsisten yonthewholeset

Ω

,one anshowuniform onsisten yon

asequen eofsu ientlylarge subsets

Ω

n

of

Ω

anddealwiththeos illationoftheestimator.

Firstnote that,sin e

Ω

is a ompa tsubset of

R

d

,foraxed

η > 1/β

f

andevery

n ∈ N \ {0}

,there

exists anite subset

Ω

n

of

Ω

with

card(Ω

n

) = O(n

c

₎

,

c > 0

su h that for all

x ∈ Ω

, one an nd

χ(x) ∈ Ω

n

satisfying

kx − χ(x)k < n

−η

. Thetriangularinequalityyields:

p

n

≤ I



sup

x∈Ω

|γ(x) − γ(χ(x))| > ε/3



+ P



sup

ω∈Ω

n

|bγ

a

(ω, k

ω

, h) − γ(ω)| > ε/3



+ P



sup

x∈Ω



bγ

a

(x, k

x

, h) − bγ

a

(χ(x), k

χ(x)

, h)



 > ε/3



.

(8)

Theproofoftheuniformweak onsisten yofourestimator onsistsinshowingthatthethreetermsin

theaboveinequality onvergeto0as

n

goestoinnity. Thisis arriedoutinPropositions1,2and3.

Theorem 1is thus adire t onsequen eof these results. Westart by fo using onthe onvergen e of

therstterm.

Proposition1. Under model(

M

) and(A1), for

n

largeenough,

sup

x∈Ω

|γ(x) − γ(χ(x))| ≤ ε/3.

Proof of Proposition 1

−

Re all that forall

x ∈ Ω

,

kx − χ(x)k < n

−η

_{→ 0}

. Sin e

Ω

is ompa t,

(A1)entailsthat thefun tion

γ

isuniformly ontinuous,whi hshowstheresult.

We are now interested in the se ond term, namely in the uniform onvergen e of our estimator on

thenitesubsets

Ω

n

of

Ω

. Somepreliminarylemmasarerequired,whoseproofsare postponedtothe

Appendix. Therstoneis ausefulresultofrealanalysis.

Lemma 1. Let

(a

1

, . . . , a

n

)

and

(b

1

, . . . , b

n

)

be two

n−

tuples of pairwise distin t real numbers su h

thatfor all

i ∈ {1, . . . , n}

,

a

i

≤ b

i

. Letfurther

a

1,n

≤ . . . ≤ a

n,n

and

b

1,n

≤ . . . ≤ b

n,n

betheasso iated

ordered

n

tuples. Then for all

i ∈ {1, . . . , n}

,

a

i,n

≤ b

i,n

.

Lemma 2isatopologi alresultwhi hshallbeneededin severalproofs: itessentiallyimpliesthat for

n

largeenough,theball

B(x, h)

is ontainedin

S

forall

x ∈ Ω

.

Lemma 3belowgivesan asymptoti uniform estimation of the total numberof ovariates

M

n

(ω, h)

ontainedintheballswith enter

ω ∈ Ω

n

andradius

h

.

Lemma 3. Under model (

M

), assume that

(A1)

holds together with

nh

d

_{/ log n → ∞}

. Then, as

n

goestoinnity,

1

nh

d

sup

ω∈Ω

n



M

n

(ω, h) − Vnh

d

f (ω)





_{−→ 0.}

P

Given

M

n

(x, h) ≥ 1

, for

i = 1, . . . , M

n

(x, h)

, let

Z

(x)

i

be the response variable whose asso iated

ovariate

W

(x)

i

belongs totheball

B(x, h)

. Letusalsointrodu ethenotations

U

(x)

i

:= F (Z

(x)

i

|W

(x)

i

)

for

i = 1, . . . , M

n

(x, h)

and

V

i

= F (Y

i

|X

i

)

for

i = 1, . . . , n

. Inthefollowing,

Ω

e

denotesanite subset

of

Ω

,

m := (m

ω

)

ω∈e

Ω

isalistof positiveintegersand

B

e

Ω

(m)

istheBorelmeasurable set

B

Ω

e

(m) :=

\

ω∈e

Ω

{M

n

(ω, h) = m

ω

}.

Thedistributionsof

U

(x)

i

and

V

i

aregiveninthefollowingresult.

Lemma 4. Under model (

M

), the random variables

V

1

, . . . , V

n

are independent standard uniform

randomvariableswhi hareindependentfrom

X

1

, . . . , X

n

. Furthermore,forall

ω ∈ e

Ω

and onditionally

to

B

e

Ω

(m)

,the random variables

U

(ω)

1

, . . . , U

(ω)

m

ω

areindependent standard uniformrandom variables.

The next lemma provides arepresentation of ourestimator in terms of independent standard

expo-nentialrandomvariables, whi histhekeyargumenttoshowProposition 2.

Lemma 5. Under model (

M

) and (A3), for all

ω ∈ e

Ω

and onditionally to

B

e

Ω

(m)

, there exist

independentstandardexponentialrandomvariables

E

(ω)

1

, . . . , E

(ω)

m

ω

su hthatforeverysequen eof

real-valued fun tions

(a

n

)

denedon

Ω

su h that

a

n

(x) → a ∈ (0, 1)

uniformly in

x ∈ Ω

, one has for

n

largeenough, uniformlyin

ω ∈ e

Ω

,





bγ

a

n

(ω)

(ω, k

ω

, h) − γ(ω)E

(ω)

n





 ≤ 2ω(U

1,m

(ω)

ω

, U

(ω)

m

ω

,m

ω

, ω, h) + E

(ω)

n

∆

ω

(U

_k

(ω)

_ω

_,m

_ω

)

≤ 2ω(V

1,n

, V

n,n

, ω, h) + E

(ω)

n

∆

ω

(U

_k

(ω)

_ω

_,m

_ω

)

where

E

(ω)

n

:=

1

k

ω

− k

ω,a

n

(ω)

+ 1

k

ω

X

l=k

ω,an (ω)

1

l − 1

l−1

X

i=1

E

_i

(ω)

.

Wearenowin positionto provetheuniform onsisten yofourestimatoronthenitesubsets

Ω

n

.

Proposition2. Under model(

M

),assume that(A1), (A2) and(A3)hold. If

nh

d

_{/ log n → ∞}

,

inf

x∈Ω

min



k

x

log n

,

nh

d

k

x

log(nh

d

)



→ ∞

and

lim

t→0

sup

_x∈Ω

∆

x

(t) = 0,

then, for every sequen e of real-valued fun tions

(a

n

)

dened on

Ω

su h that

a

n

(x) → a ∈ (0, 1)

uniformlyin

x ∈ Ω

as

n

goestoinnity,

sup

ω∈Ω

n

|bγ

a

n

(ω)

(ω, k

ω

, h) − γ(ω)|

P

−→ 0.

Proposition 2with the onstantsequen e

a

n

= a

for all

n ≥ 1

. Proposition 2 also handlesthe ase

when

(a

n

)

isanarbitrarysequen eofreal-valuedfun tionson

Ω

uniformly onvergingto

a

,whi hshall

beusefulto establishProposition3.

Proof ofProposition2

−

Let

m = (m

ω

)

ω∈Ω

n

bealistofpositiveintegerssu hthat

∀ω ∈ Ω

n

,

m

ω

f (ω)nh

d

∈



_V

2

,

3V

2



,

(9)

andlet

L

n

bethesetofallpossiblelistssatisfying(9). FromLemma3,itis learthat

P

(A

n

) → 1

as

n

goestoinnity,where

A

n

:=

[

m∈L

n

B

Ω

n

(m)

isthedisjointunionofthe

B

Ω

n

(m)

for

m ∈ L

n

. Let

ε > 0

. Remarkingthat

P



sup

ω∈Ω

n

|bγ

a

n

(ω)

(ω, k

ω

, h) − γ(ω)| > ε



≤ P(A

C

n

) +

X

m∈L

n

P



sup

ω∈Ω

n

|bγ

a

n

(ω)

(ω, k

ω

, h) − γ(ω)| > ε; B

Ω

n

(m)



≤ P(A

C

n

) + sup

m∈L

n

P



sup

ω∈Ω

n

|bγ

a

n

(ω)

(ω, k

ω

, h) − γ(ω)| > ε







 B

Ω

n

(m)



,

where

A

C

n

is the omplementof

A

n

,itissu ienttoprovethatas

n

goestoinnity,

sup

m∈L

n

T (m) := sup

m∈L

n

P



sup

ω∈Ω

n

|bγ

a

n

(ω)

(ω, k

ω

, h) − γ(ω)| > ε







 B

Ω

n

(m)



→ 0.

(10)

Let

m ∈ L

n

. Remarking that

T (m) ≤ P



sup

ω∈Ω

n

|bγ

a

n

(ω)

(ω, k

ω

, h) − γ(ω)E

(ω)

n

| >

ε

2







 B

Ω

n

(m)



+P



sup

ω∈Ω

n

|γ(ω)(E

(ω)

n

− 1)| >

ε

2







 B

Ω

n

(m)



,

wehavefromLemmas4and5that

T (m) ≤ P



sup

ω∈Ω

n

ω(V

1,n

, V

n,n

, ω, h) >

ε

8



+ P



sup

ω∈Ω

n

E

(ω)

n

∆

ω

(U

k

(ω)

ω

,m

ω

) >

ε

4







 B

Ω

n

(m)



+ P



sup

ω∈Ω

n

|γ(ω)(E

(ω)

n

− 1)| >

ε

2







 B

Ω

n

(m)



≤ P



sup

ω∈Ω

n

ω(V

1,n

, V

n,n

, ω, h) >

ε

8



+ card(Ω

n

)



sup

ω∈Ω

n

P



_|γ(ω)(E

(ω)

n

− 1)| >

ε

2





 B

Ω

n

(m)



+

sup

ω∈Ω

n

P



E

(ω)

n

∆

ω

(U

k

(ω)

ω

,m

ω

) >

ε

4





 B

Ω

n

(m)



=: T

1

(m) + card(Ω

n

)(T

2

(m) + T

3

(m)).

First,letus onsidertheterm

T

1

(m)

. Under ondition(A2),for

n

largeenoughanduniformlyin

m

,

T

1

(m) ≤ P(V

1,n

< n

−(1+δ)

) + P(V

n,n

> 1 − n

−(1+δ)

) = 2(1 − (1 − n

−(1+δ)

)

n

) → 0.

(11)

Regarding

T

2

(m)

,itiseasytoseethat for

n

largeenough

T

2

(m) ≤ sup

ω∈Ω

n

k

ω

X

l=k

_{ω,an (ω)}

P









1

l − 1

l−1

X

i=1

γ(ω)(E

i

(ω)

− 1)











>

ε

2











B

Ω

n

(m)

!

.

ablestogetherwith(A1),thereexistsapositive onstant

C

ε

su hthat,for

n

largeenough,

T

2

(m) ≤ 2 sup

ω∈Ω

n

k

ω

X

l=k

ω,an(ω)

exp (−C

ε

(l − 1)) ≤ 2 exp



−

C

ε

2

ω∈Ω

inf

n

(k

ω,a

n

(ω)

− 1)



.

Finally,using thefa tthat

card(Ω

n

) = O(n

c

₎

,

k

x,a

n

(x)

/k

x

→ 1 − a

and

k

x

/ log(n) → ∞

uniformlyin

x ∈ Ω

,onehas,for

n

su ientlylarge,uniformly in

m

,

card(Ω

n

)T

2

(m) ≤ 2 exp



−

C

ε

4

ω∈Ω

inf

n

(k

ω,a

n

(ω)

− 1)



→ 0.

(12)

Wenowfo uson

T

3

(m)

. Letusdene

ε

2

n

= sup

x∈Ω

k

x

log(nh

d

)

nh

d

.

Clearly,

ε

n

→ 0

as

n

goestoinnityand

T

3

(m) ≤ sup

ω∈Ω

n

P



_U

(ω)

k

ω

,m

ω

> ε

n





 B

Ω

n

(m)



+ sup

ω∈Ω

n

P



E

(ω)

n

sup

x∈Ω

∆

x

(ε

n

) >

ε

4







 B

Ω

n

(m)



.

UsingLemma 4,wehave:

P



_U

(ω)

k

ω

,m

ω

> ε

n





 B

Ω

n

(m)



=

m

ω

!

(k

ω

− 1)!(m

ω

− k

ω

)!

Z

1

ε

n

x

k

ω

−1

_{(1 − x)}

m

ω

−k

ω

_dx

≤ m

k

ω

ω

(1 − ε

n

)

m

ω

−k

ω

.

Remarking that

log(1 − ε

n

) < −ε

n

/2

for

n

largeenough,onehas,forall

m

and

ω

,

P



U

_k

(ω)

_ω

_,m

_ω

> ε

n





 B

Ω

n

(m)



≤ exp



−m

ω

ε

n



m

ω

− k

ω

2m

ω

−

k

ω

log m

ω

m

ω

ε

n



.

Furthermore,under(A1),sin e

m

satises(9),wehave:

k

ω

m

ω

≤

2

f V

ε

2

n

log(nh

d

₎

and

log(m

ω

) ≤ log



3f

V

2

nh

d



≤

3

2

log(nh

d

_),

forall

m

and

ω

. Thus, for

n

su ientlylarge,uniformlyin

m

and

ω

,

P



_U

(ω)

k

ω

,m

ω

> ε

n





 B

Ω

n

(m)



≤ exp



−

1

4

ω∈Ω

inf

n

m

ω

ε

n



≤ exp



−

f V

8

nh

d

_ε

n



.

Furthermore,sin e

log n/(nh

d

_ε

n

) → 0

and

card(Ω

n

) = O(n

c

₎

, it is straightforward that for

n

su- ientlylarge,uniformlyin

m

,

card(Ω

n

) sup

ω∈Ω

n

P



_U

(ω)

k

ω

,m

ω

> ε

n





 B

Ω

n

(m)



≤ exp



−

f V

16

nh

d

_ε

n



.

(13)

Next,sin e

ε

n

→ 0

andthat,byassumption,for

n

largeenough,

sup

x∈Ω

∆

x

(ε

n

) ≤

ε

8

,

onehas,under(A1):

sup

ω∈Ω

n

P



E

(ω)

n

sup

x∈Ω

∆

x

(ε

n

) >

ε

4







 B

Ω

n

(m)



≤

sup

ω∈Ω

n

P



sup

x∈Ω

∆

x

(ε

n

)|E

(ω)

n

− 1| >

ε

8







 B

Ω

n

(m)



≤

sup

ω∈Ω

n

P



γ(ω)|E

(ω)

n

− 1| > γ





 B

Ω

n

(m)



.

Therighthand-sideoftheaboveinequalityissimilarto

T

2

(m)

andthus(12)and(13)leadto

card(Ω

n

)T

3

(m) ≤ exp



−

f V

16

nh

d

_ε

n



+ 2 exp



−

C

′

ε

4

ω∈Ω

inf

n

(k

ω,a

n

(ω)

− 1)



→ 0,

(14)

for

n

large enough, uniformly in

m

, where

C

′

ε

is a positive onstant. We then easily obtain (10)

using(11),(12)and(14)andtheproofis omplete.

The os illationof the fun tion

x 7→ bγ

a

(x, k

x

, h)

isstudied in Proposition 3. Theproofof this result

requires to ontroltherandomvariable

C

h

(x, r) =

n

X

i=1

I

{h − r ≤ kX

i

− xk ≤ h + r},

whi histhetotalnumberof ovariatesintheannuluswith enter

x

,innerradius

h− r

andouterradius

h + r

. Lemma 6belowessentiallystatesthat this numberisasymptoti allyboundedwith arbitrarily

largeprobability.

Lemma 6. Under model (

M

), assume that (A1) holds together with

nh

d

_{→ ∞}

. Then, for every

arbitrary integer

K

2

> c/(ηβ

f

− 1)

,

P

(A

n,K

2

) → 0

as

n

goes toinnity,where

A

n,K

2

=



sup

ω∈Ω

n

C

h

(ω, n

−η

) ≥ K

2



.

We annowstateandproveProposition3: theos illationofthefun tion

x 7→ bγ

a

(x, k

x

, h)

onverges

uniformlyto 0inprobability.

Proposition3. Under model(

M

),assume that(A1), (A2) and(A3)hold. If

nh

d

_{/ log n → ∞}

,

inf

x∈Ω

min



k

x

log n

,

nh

d

k

x

log(nh

d

)



→ ∞, lim

t→0

sup

_x∈Ω

∆

x

(t) = 0,

andif thereexistsanite positive onstant

K

1

su hthat

sup

x∈Ω

sup

kx

′

−xk≤h

|k

x

− k

x

′

| ≤ K

1

,

then, as

n

goestoinnity, if

a ∈ (0, 1)

,

sup

x∈Ω



bγ

a

(x, k

x

, h) − bγ

a

(χ(x), k

χ(x)

, h)





_{−→ 0.}

P

Proof of Proposition 3

−

FromLemma 6, itis enough toshowthat for all

ε > 0

and foraxed

integer

K

2

> c/(ηβ

f

− 1)

,

P



sup

x∈Ω



bγ

a

(x, k

x

, h) − bγ

a

(χ(x), k

χ(x)

, h)



 > ε







 A

C

n,K

2



→ 0.

For

(k, l) ∈ {2, . . . , n}

2

and

i ∈ {1, . . . , n − 1}

,let

r

i,l

(x, k, h) =

I

_{kX

∗

i

− xk ≤ h; M

l

(x, h) ≥ k}

M

l

(x, h) − 1

,

if

M

l

(x, h) > 1

and0elsewhere,and,for

a ∈ (0, 1)

and

k

a

= ⌊(1 − a)k⌋

,

s

l,a

(x, k, h) =

I

{kX

∗

l

− xk ≤ h; M

l

(x, h) ≤ k}

k − k

a

+ 1

bγ

a

(x, k

x

, h) =

n

X

l=2

l−1

X

i=1

log

Y

n−i+1,n

Y

n−l+1,n

r

i,l

(x, k

x,a

, h)s

l,a

(x, k

x

, h),

andthus



bγ

a

(x, k

x

, h) − bγ

a

(χ(x), k

χ(x)

, h)



 ≤ S

n,1

(x) + S

n,2

(x)

,where

S

n,1

(x)

:=

n

X

l=2

l−1

X

i=1

log

Y

n−i+1,n

Y

n−l+1,n

|r

i,l

(x, k

x,a

, h) − r

i,l

(χ(x), k

χ(x),a

, h)|s

l,a

(x, k

x

, h),

and

S

n,2

(x)

:=

n

X

l=2

l−1

X

i=1

log

Y

n−i+1,n

Y

n−l+1,n

|s

l,a

(x, k

x

, h) − s

l,a

(χ(x), k

χ(x)

, h)|r

i,l

(χ(x), k

χ(x),a

, h).

Theideaoftherestoftheproofisquitesimple. Wewillshowthatontheevent

A

C

n,K

2

,thereexisttwo

sequen esof real-valuedfun tions

(a

−

n

)

and

(a

+

n

)

on

Ω

uniformly tendingto

a

, four sequen es

(α

−

1,n

)

,

(α

+

1,n

)

,

(α

−

2,n

)

and

(α

+

2,n

)

tendingto1and apositive onstant

K

3

su h that,forall

x ∈ Ω

S

n,1

(x) ≤ 2



α

+

_1,n

bγ

a

+

n

(χ(x))

χ(x), k

χ(x)

+ K

3

, h

+

_{− α}

−

1,n

bγ

a

−

n

(χ(x))

χ(x), k

χ(x)

− K

3

, h

−



_,

(15) and

S

n,2

(x) ≤ 2



α

+

2,n

bγ

a

+

n

(χ(x))

χ(x), k

χ(x)

+ K

3

, h

+

_{− α}

−

2,n

bγ

a

−

n

(χ(x))

χ(x), k

χ(x)

− K

3

, h

−



(16) where

h

±

_{:= h ± n}

−η

. Sin e

inf

x∈Ω

k

x

→ ∞

,

h

±

_{= h(1 + o(1))}

and thefun tion

γ

is bounded from

belowandabovebypositive onstants,adire tuseofProposition2shallthenleadto

sup

x∈Ω

S

n,1

(x)

P

−→ 0

and

sup

x∈Ω

S

n,2

(x)

P

−→ 0,

whi hwillthen on ludetheproofofProposition3. Toobtain(15)and(16),thefollowing

straightfor-wardresultswillbeuseful. Forall

(x, x

′

_{) ∈ Ω}

2

su hthat

kx − x

′

_{k ≤ n}

−η

andforall

i ∈ {1, . . . , n − 1}

,

|I{kX

i

∗

− xk ≤ h} − I{kX

i

∗

− x

′

k ≤ h}| ≤ I{h

−

≤ kX

i

∗

− x

′

k ≤ h

+

}.

(17)

Furthermore,fromtheinequalities

|M

l

(x, h) − M

l

(x

′

, h)| ≤ C

h

(x

′

, n

−η

)

and

|M

l

(x

′

_{, h) − M}

l

(x

′

, h

±

)| ≤ C

h

(x

′

, n

−η

),

thetriangularinequalityyields,forall

l ∈ {2, . . . , n}

,on

A

C

n,K

2

,



M

l

(x, h) − M

l

(x

′

, h

±

)



 ≤ 2C

h

(x

′

, n

−η

) ≤ 2K

2

.

(18) Espe ially,if

M

l

(x, h) > 1

andon

A

C

n,K

2

,

M

l

(x

′

, h

+

) − 1

M

l

(x, h) − 1

≤ 1 +

2K

2

M

l

(x, h) − 1

and

M

l

(x

′

, h

−

) − 1

M

l

(x, h) − 1

≥ 1 −

2K

2

M

l

(x, h) − 1

.

(19)

Letusrstfo usontheterm

S

n,1

(x)

. Itiseasyto seethat

D

_i,l

(r)

(x, a, h) := |r

i,l

(x, k

x,a

, h) − r

i,l

(χ(x), k

χ(x),a

, h)| ≤ T

(r)

n,1

(x) + T

(r)

n,2

(x) + T

(r)

n,3

(x),

T

n,1

(r)

(x) =

|I{kX

∗

i

− xk ≤ h} − I{kX

i

∗

− χ(x)k ≤ h}|I{M

l

(χ(x), h) ≥ k

χ(x),a

}

M

l

(x, h) − 1

,

if

M

l

(χ(x), h) ≥ k

χ(x),a

and 0otherwise,

T

_n,2

(r)

(x) =

|I{M

l

(x, h) ≥ k

x,a

} − I{M

l

(χ(x), h) ≥ k

χ(x),a

}|I{kX

∗

i

− xk ≤ h}

M

l

(x, h) − 1

,

if

M

l

(x, h) ≥ k

x,a

or

M

l

(χ(x), h) ≥ k

χ(x),a

and0otherwiseand

T

n,3

(r)

(x) =

|M

l

(χ(x), h) − M

l

(x, h)|I{kX

i

∗

− χ(x)k ≤ h}I{M

l

(χ(x), h) ≥ k

χ(x),a

}

(M

l

(x, h) − 1)(M

l

(χ(x), h) − 1)

,

if

M

l

(χ(x), h) ≥ k

χ(x),a

and 0 otherwise. Note that for

n

large enough, sin e

inf

x∈Ω

k

x,a

→ ∞

and (18)holds, if

M

l

(χ(x), h) ≥ k

χ(x),a

then

M

l

(x, h) > 1

andthusthe terms

T

(r)

n,i

(x)

,

i = 1, 2, 3

are

asymptoti allywelldened. Wenowstudyseparately thesethree terms. For

u ∈ R

, letusintrodu e

thequantities

ξ

+

(u) = sup

x∈Ω



1 +

2K

2

k

x,a

− u − 1



, ξ

−

(u) = inf

x∈Ω



1 −

2K

2

k

x,a

− u − 1



,

ζ

+

(u) = sup

x∈Ω

k

χ(x)

− k

χ(x),a

+ 1 + u

k

x

− k

x,a

+ 1

and

ζ

−

_{(u) = inf}

x∈Ω

k

χ(x)

− k

χ(x),a

+ 1 − u

k

x

− k

x,a

+ 1

.

Clearly,forall

u ∈ R

,

ξ

±

_(u)

and

ζ

±

_(u)

onvergetooneas

n

goestoinnity. From(17),(18)and(19),

sin eforall

l ∈ {2, . . . , n}

,

i ∈ {1, . . . , n − 1}

and

x ∈ Ω

,

r

i,l

(x, ., h)

isade reasingfun tion,onehas

T

n,1

(r)

(x)

≤

M

l

(χ(x), h

+

) − 1

M

l

(x, h) − 1

r

i,l

(χ(x), k

χ(x),a

, h

+

) −

M

l

(χ(x), h

−

) − 1

M

l

(x, h) − 1

r

i,l

(χ(x), k

χ(x),a

, h

−

)

≤ ξ

+

_(2K

2

)r

i,l

(χ(x), k

χ(x),a

+ K

4

, h

+

) − ξ

−

(2K

2

)r

i,l

(χ(x), k

χ(x),a

− K

4

, h

−

),

(20)

where

K

4

= (a − 1)K

1

− 2K

2

− 1

. Similarly,sin e

|k

x,a

− k

χ(x),a

| ≤ (1 − a)K

1

+ 1

uniformlyin

x ∈ Ω

,

notingthat

|I{M

l

(x, h) ≥ k

x,a

} − I{M

l

(χ(x), h) ≥ k

χ(x),a

}| ≤ I{k

χ(x),a

+ K

4

≤ M

l

(χ(x), h) < k

χ(x),a

− K

4

}

yields

T

n,2

(r)

(x)

≤

M

l

(χ(x), h

+

) − 1

M

l

(x, h) − 1

r

i,l

(χ(x), k

χ(x),a

+ K

4

, h

+

) −

M

l

(χ(x), h

−

_{) − 1}

M

l

(x, h) − 1

r

i,l

(χ(x), k

χ(x),a

− K

4

, h

−

)

≤ ξ

+

(2K

2

− K

4

)r

i,l

(χ(x), k

χ(x),a

+ K

4

, h

+

) − ξ

−

(2K

2

+ K

4

)r

i,l

(χ(x), k

χ(x),a

− K

4

, h

−

).

(21)

Clearly

T

n,3

(r)

(x) ≤

K

2

ξ

+

(2K

2

)

k

χ(x),a

− 1

r

i,l

(χ(x), k

χ(x),a

+ K

4

, h

+

),

(22) and

K

2

ξ

+

_(2K

2

)/(k

χ(x),a

− 1) → 0

uniformly in

x ∈ Ω

. Furthermore, using on e again (17), (18)

and(19),letting

K

3

= K

1

+ 2K

2

and

K

5

= K

3

− K

4

,onehas

ζ

−

(K

5

)s

_l,a

−

n

(χ(x))

(χ(x), k

χ(x)

−K

3

, h

−

_{) ≤ s}

l,a

(x, k

x

, h) ≤ ζ

+

(K

5

)s

_l,a

+

n

(χ(x))

(χ(x), k

χ(x)

+K

3

, h

+

_),

(23)

wherethesequen esoffun tions

(a

+

n

)

and

(a

−

n

)

aregivenby

∀ x ∈ Ω, a

±

n

(x) = 1 −

k

x,a

± K

4

k

x

± K

3

.

Colle ting(20)to (23)itiseasyto onstru ttwosequen es

(α

−

1,n

)

and

(α

+

1,n

)

tendingto1su hthat

D

(r)

i,l

(x, a, h)s

l,a

(x, k

x

, h) ≤ 2



α

+

1,n

r

i,l

(χ(x), k

χ(x),a

+ K

4

, h

+

)s

_l,a

+

n

(χ(x))

(χ(x), k

χ(x)

+ K

3

, h

+

₎

− α

−

_1,n

r

i,l

(χ(x), k

χ(x),a

− K

4

, h

−

)s

_l,a

−

n

(χ(x))

(χ(x), k

χ(x)

− K

3

, h

−

₎



_,

whi h on ludestheproofof(15). Wenowturn to

S

n,2

(x)

. Werststartfromthede omposition

D

_l,a

(s)

(x, k

x

, h) := |s

l,a

(x, k

x

, h) − s

l,a

(χ(x), k

χ(x)

, h)| ≤ T

n,1

(s)

(x) + T

(s)

n,2

(x) + T

(s)

n,3

(x),

where

T

_n,1

(s)

(x) =

|I{M

l

(x, h) ≤ k

x

} − I{M

l

(χ(x), h) ≤ k

χ(x)

}|I{kX

∗

l

− χ(x)k ≤ h}

k

χ(x)

− k

χ(x),a

+ 1

,

T

n,2

(s)

(x) =

|I{kX

∗

l

− xk ≤ h} − I{kX

l

∗

− χ(x)k ≤ h}|I{M

l

(x, h) ≤ k

x

}

k

χ(x)

− k

χ(x),a

+ 1

,

and

T

n,3

(s)

(x) =









1

k

x

− k

x,a

+ 1

−

1

k

χ(x)

− k

χ(x),a

+ 1









I

{kX

l

∗

− xk ≤ h}I{M

l

(x, h) ≤ k

x

}.

A onjointuseof(17),(18)and(19)leadsto

T

n,1

(s)

(x) ≤

I

{kX

∗

l

− χ(x)k ≤ h

+

}I{M

l

(χ(x), h

+

) ≤ k

χ(x)

+ K

3

}

k

χ(x)

− k

χ(x),a

+ 1

−

I

{kX

∗

l

− χ(x)k ≤ h

−

}I{M

l

(χ(x), h

−

) ≤ k

χ(x)

− K

3

}

k

χ(x)

− k

χ(x),a

+ 1

≤

ζ

+

_(K

5

)

ζ

−

₍₀₎

s

l,a

+

n

(χ(x))

(χ(x), k

χ(x)

+ K

3

, h

+

_{) −}

ζ

−

(K

5

)

ζ

+

₍₀₎

s

l,a

−

n

(χ(x))

(χ(x), k

χ(x)

− K

3

, h

−

_).

(24) Similarly,

T

_n,2

(s)

(x) ≤

I

{h

−

_{≤ kX}

∗

l

− χ(x)k ≤ h

+

}

k

χ(x)

− k

χ(x),a

+ 1

I

_{M

_l

_{(x, h) ≤ k}

_x

_}

≤

ζ

+

_(K

5

)

ζ

−

₍₀₎

s

l,a

+

n

(χ(x))

(χ(x), k

χ(x)

+ K

3

, h

+

_{) −}

ζ

−

(K

5

)

ζ

+

₍₀₎

s

l,a

−

n

(χ(x))

(χ(x), k

χ(x)

− K

3

, h

−

_).

(25) Next,(23)yields

T

n,3

(s)

(x)

≤



(ζ

+

(0) − 1) ∨ (1 − ζ

−

(0))



k

x

− k

x,a

+ 1

k

χ(x)

− k

χ(x),a

+ 1

s

l,a

(x, k

x

, h)

≤



(ζ

+

(0) − 1) ∨ (1 − ζ

−

(0))

 ζ

+

_(K

5

)

ζ

−

₍₀₎

s

l,a

+

n

(χ(x))

(χ(x), k

χ(x)

+ K

3

, h

+

_).

(26) Remarking that

ξ

−

(0)r

i,l

(χ(x), k

χ(x),a

− K

4

, h

−

) ≤ r

i,l

(χ(x), k

χ(x),a

, h) ≤ ξ

+

(0)r

i,l

(χ(x), k

χ(x),a

+ K

4

, h

+

),

(27)

and olle ting(24)to(27),one anndsequen es

(α

−

2,n

)

and

(α

+

2,n

)

tendingto 1su h that

D

(s)

_l,a

(x, k

x

, h)r

i,l

(χ(x), k

χ(x),a

, h) ≤ 2



α

+

_2,n

r

i,l

(χ(x), k

χ(x),a

+ K

4

, h

+

)s

_l,a

+

n

(χ(x))

(χ(x), k

χ(x)

+ K

3

, h

+

₎

− α

−

_2,n

r

i,l

(χ(x), k

χ(x),a

− K

4

, h

−

)s

l,a

−

n

(χ(x))

(χ(x), k

χ(x)

− K

3

, h

−

₎



_,