Estimating an endpoint with high order moments Stéphane Girard, Armelle Guillou, Gilles Stupfler

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Estimating an endpoint with high order moments

Stéphane Girard, Armelle Guillou, Gilles Stupfler

To cite this version:

Stéphane Girard, Armelle Guillou, Gilles Stupfler. Estimating an endpoint with high order moments. Test, Spanish Society of Statistics and Operations Research/Springer, 2012, 21 (4), pp.697-729.

�10.1007/s11749-011-0277-8�. �inria-00596979�

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Stéphane Girard

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, Armelle Guillou

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& GillesStuper

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TeamMistis,INRIARhône-Alpes&LJK,Inovallée,655, av. del'Europe,

Montbonnot,38334 Saint-Ismiercedex,France

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UniversitédeStrasbourg&CNRS,IRMA,UMR7501,7rueRenéDescartes,

67084Strasbourgcedex,France

Abstract. Wepresentanewmethodforestimatingtheendpointofaunidimensionalsample

whenthedistributionfunctiondecreasesatapolynomialratetozerointheneighborhoodofthe

endpoint. Theestimatoris basedon theuseofhigh ordermomentsofthevariable ofinterest.

Itis assumedthat theorderofthemomentsgoesto innity,andwegiveconditionsonitsrate

of divergence to getthe asymptotic normalityof the estimator. The good performanceof the

estimatorisillustratedonsomenite samplesituations.

AMSSubject Classications: 62G32,62G05.

Keywords: Endpointestimation,high ordermoments,consistency,asymptoticnormality.

1 Introduction

Let

(X 1 , . . . , X n )

^be

n

independentcopiesofarandomvariable

X

^,^with^bounded^support

[0, θ]

^,

where

θ > 0

îs ûnknown. În ^this ^paper, ^we âddress ^the ^problem ôf êstimating ^the ^(right)

endpoint

θ

^of ^the ^survival^function

F

^of

X

^. ^Pioneering ^work ôn êndpoint êstimation încludes

Quenouille (1949) who introduced a jackknife estimate of the endpoint based on the naive

maximumestimator. ThisapproachwasfurtherstudiedbyMiller(1964),RobsonandWhitlock

(1964),Cooke(1979)anddeHaan(1981),tonameafew. Awell-knownreferenceonendpoint

estimation is Hall (1982), recently improved by Li and Peng (2009), in which a maximum

likelihoodmethod isusedwhen

F

^belongs^to ^the^Hall^model, ^see^for^instance ^Section^5. ^Hall's

work gave a start to the study of general threshold-based methods, together with the use of

theapproximationof excessesbyGeneralizedParetoDistributions,see forinstance Smithand

Weissman(1985)andSmith(1987). Ageneralconstructionofestimatorsoftheendpointusing

athreshold is givenin deHaan andFerreira(2006, p. 146). Somepopularestimators in this

framework,called Peaks OverThreshold (POT) approach, are probability weightedmoments

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likelihoodestimators(Dreesetal.,2003).

Other studies include Loh (1984) and Athreyaand Fukuchi (1997) with abootstrapmethod,

Halland Wang(1999)foraminimal-distance method, Goldenshlugerand Tsybakov(2004)for

endpoint estimation in presence of random errors, and Hall and Wang (2005) for a Bayesian

likelihoodapproach. Asfarasdetectingthepresenceofaniteendpointisconcerned,seeNeves

andPereira(2010).

Inthispaper,weintroduceanestimatorusinghighmomentsofthevariableofinterest

X

^. ^More

precisely,theestimatorisgivenby

1 b θ n

= 1 ap n

((a + 1)p n + 1) b µ (a+1)p n

b

µ (a+1)p n +1

− (p n + 1) µ b p n

b µ p n +1

(1)

where

(p n )

^is^a^nonrandom^sequence^such ^that

p n → ∞

^,

a > 0

^and

b

µ p n = 1 n

X n i=1

X _i ^p ⁿ

is the classicalmoment estimator of

µ p n := E (X ^p ⁿ )

^. ^From ^a ^practical ^point ^of ^view, ^taking

high order momentsgives moreweight to observations closeto

θ

^. ^From^a theoreticalpointof view,theestimator

θ b n

^convergesⁱⁿprobabilityto

θ

^withoutâny^parametricâssumptionôn^the

distribution of

X

^, ^see ^Section ^3. ^Theâsymptotic ^normalityôf ^theêstimator îs established in Section 4under asemi-parametric assumption. Some examplesof distributions satisfyingthis

assumptionareprovidedin Section5. Somesimulationsareproposed inSection 6to illustrate

the eciency of ourestimator, and to compare it with estimators of the endpoint estimation

literature. Auxiliaryresultsarepostponedto AppendixAandprovedinAppendixB.

2 Construction of the estimator

Tojustifytheintroductionofourestimator(1), letrst

Y

^be^a^random^variable ^with^survival

function

G

^dened ^by

G(y) = (1 − y/θ) ^α

^for^all

y ∈ [0, θ]

^. ^We^get^for^all

p ≥ 1

^,

M p := E (Y ^p ) = p

Z +∞

0 y ^p−1 G(y) dy = α θ ^p B(p + 1, α)

⁽²⁾

where

B(x, y) = Z 1

0 t ^x−1 (1 − t) ^y−1 dt

^is^the^Beta^function. ^This^yields ^for^all

p ≥ 1

^,

M p

M p+1 = 1 θ

1 + α

p + 1

(3)

leadingto,forallarbitrarysequences

(p n )

^and^all

a > 0 1

θ = 1 ap n

((a + 1)p n + 1) M (a+1)p n

M (a+1)p n +1

− (p n + 1) M p n

M p n +1

.

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Usingtheaboveideas,weshallthendeneourestimatorintwosteps. First,themoment

M p

^is

replacedbythetruemoment

µ p

^,^and^we^set

1 Θ n

:= 1 ap n

((a + 1)p n + 1) µ (a+1)p n

µ (a+1)p n +1

− (p n + 1) µ p n

µ p n +1

.

Second,

µ p n

^is ^estimated ^by ^the corresponding empirical moment

µ b p n

^; ^plugging

µ b p n

ⁱⁿ

1/Θ n

yieldstheestimator(1)of

1/θ

^.

3 Consistency

Inthissection,westateandprovetheconsistencyofourestimatorinanon-parametriccontext.

Theonlyhypothesisis

(A 0 ) X

îs^positiveând^theêndpoint

θ = sup{x ≥ 0 | F (x) < 1}

^of

X

^is^nite.

Tothisend, therststepistoprovearesultsimilarto(3)for

µ p n

^.

Proposition1. Under

(A 0 )

^,

µ p n /µ p n +1 −→ 1/θ

^as

n → ∞

^.

This resultisastraightforwardconsequenceofLemma 1. Thesecond stepconsistsin showing

that

µ p n

^can ^be ^replaced ^by ^its ^empirical counterpart

µ b p n

^. ^Dening

µ 1, p n = µ p n /θ ^p ⁿ

^as ⁱⁿ

Appendix A,wehavethefollowingresult:

Proposition2. Assumethat

(A 0 )

^holds. ^If

n µ 1, p n −→ ∞

^,^then

µ b p n /µ p n

−→ P 1

^as

n → ∞

^.

Proof. Let

Y nj := [X j /θ] ^p ⁿ

^and

Z nj := Y nj /(n µ 1, p n )

^for

1 ≤ j ≤ n

^. ^The ^desired ^result ^is

thentantamountto

P n

j=1 Z nj → 1

ⁱⁿprobability. Noticenextthatforall

n

^,^the

(Z nj ) 1≤j≤n

^are

positiveindependentrandomvariables,and

P n

j=1 E (Z nj ) = 1

^. Âccording^to^Chowând ^Têicher

(1997,Corollary2p. 358),itisenoughto showthat

∀ ε > 0,

X n j=1

E (Z nj 1l {Z nj ≥ε} ) → 0

as

n → ∞

^. ^The

(Z nj ) 1≤j≤n

^beingidenticallydistributed,itisequivalenttoprovethat

∀ ε > 0, 1 µ 1, p n

E (Y n1 1l {Y n 1 ≥εn µ _{1, pn} } ) → 0.

Since

Y n1 ∈ [0, 1]

^almost^surely^and

n µ 1, p n → ∞

^,^we^get,^for^suciently^large

n 1

µ 1, p n

E (Y n1 1l {Y n 1 ≥εn µ _{1, pn} } ) = 0

andtheresultisproved.

Theorem1. Suppose

(A 0 )

^holds. ^If

n µ 1, (a+1)p n → ∞

^then

b θ n

−→ P θ

^as

n → ∞

^.

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Proof. Remark rstthat

µ 1, (a+1)p n ≤ (a + 1)µ 1, p n

^so ^that

n µ 1, p n → ∞

^. ^Second,^Lemma ¹

entails

n µ 1, p n +1 → ∞

^and

n µ 1,(a+1)p n +1 → ∞

^as

n → ∞

^. ^We^can^then ^apply Proposition 2 torewritethefrontierestimatoras

1 θ b n

= 1 ap n

((a + 1)p n + 1) µ (a+1)p n

µ (a+1)p n +1

(1 + o ^P (1)) − (p n + 1) µ p n

µ p n +1

(1 + o ^P (1))

.

UsingonceagainLemma 1yields

µ p n /µ p n +1 → 1/θ

^and

µ (a+1)p n /µ (a+1)p n +1 → 1/θ

^as

n → ∞

^.

Replacingin theaboveequality,theconclusionfollows.

4 Asymptotic normality

Wenowexaminetheasymptoticnormalityofourestimator. Tothisend,additionalassumptions

areintroduced:

(A 1 ) ∀ x ∈ [0, θ]

^,

F (x) = (1 − x/θ) ^α L((1 − x/θ) ⁻¹ )

^where

θ, α > 0

^and

L

^is^a^slowly^varying

function atinnity,i.e. suchthat

L(ty)/L(y) → 1

^as

y → ∞

^for^all

t > 0

^.

(A 2 ) ∀ x ≥ 1

^,

L(x) = exp R x

1 η(t) t ⁻¹ dt

, where

η

^is ^a^Borel^bounded^function ^tending^to

0

at innity,continuouslydierentiableon

(1, +∞)

^, ^ultimately^monotonic^and^nonidentically

0

^.

Besides,there exists

ν ≤ 0

^such^that

x η ⁰ (x)/η(x) → ν

^as

x → +∞

^.

In the general context of extreme-value theory,

(A 1 )

^entails ^that ^the distribution belongs to the Weibullmax-domain of attraction with extreme-valueindex

−1/α

^, ^we ^refer^the ^reader ^to

de Haan and Ferreira (2006). Regarding

(A 2 )

^,

L(x) = exp R x

1 η(t) t ⁻¹ dt

is the Karamata

representationfornormalizedslowlyvaryingfunctions,seeBinghametal. (1987),p.15. Under

(A 2 )

^, ^the^function

|η|

^is^ultimately non-increasingand regularlyvarying at innitywith index

ν

^,^see^Binghamêtâl. ^(1987),^paragraph^1.4.2. În^theextreme-valueframework,

ν

^is^referred^to

asthe second order parameterand

(A 2 )

îs â^second ôrder ^condition. ^Finally^, ^let ûs ^note ^that

(A 2 )

^implies^that

x η ⁰ (x) = O(η(x))

^,^so ^that

x η ⁰ (x) → 0

^as

x → +∞

^.

Werstshowthat(3)stillholds, uptoanerrorterm,when

M p

^is^replaced^by

µ p

^.

Proposition3. Assumethat

(A 1 )

^and

(A 2 )

^hold. ^Then,

µ p

µ p+1

= M p

M p+1

+ O |η(p)|

p

.

Proof. Consideringthechangeofvariables

y = (1 − x/θ) ⁻¹

ⁱⁿ ⁽²⁾^yields

M p = p ^−α θ ^p Γ(α + 1) R M (p)

with

Γ(x) = Z +∞

0 t ^x−1 e ^−t dt

^the^Gamma^function ^and

R M (p) = 1 + I 1 E 1 (p) + I 2 E 2 (p)

Γ(α + 1) ,

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where

I 1

^,

I 2

^,

E 1 (p)

^and

E 2 (p)

^are^denedⁱⁿ ^Lemma⁷^by

E 1 (p) = 1

I 1

Z 1 0

f p (x)x ^−α−2 dx − 1, I 1 =

Z +∞

1 y ^α e ^−y dy, E 2 (p) = 1

I 2

Z +∞

1 g p (x)x ^−α−2 dx − 1, I 2 = Z 1

0 y ^α e ^−y dy,

andwhere

f p

^,

g p

^are^the^functions ^introducedⁱⁿ^Lemma^6:

∀ x ∈ (0, 1], f p (x) =

1 − 1 p

−α−1

1 + 1

(p − 1)x

−α−2

1 − 1

(p − 1)x + 1 p−1

,

∀ x ∈ [1, +∞), g p (x) =

1 − 1 px

p−1

.

Similarly,thesamechangeofvariablesyields

µ p = p ^−α θ ^p L(p) Γ(α + 1) [R M (p) + R δ (p)]

⁽⁴⁾

with

R δ (p) = I 1 δ 1 (p) + I 2 δ 2 (p) Γ(α + 1)

where

δ 1 (p)

^and

δ 2 (p)

^are^dened ⁱⁿ^Lemma ⁷^by

δ 1 (p) = 1

I 1

Z 1 0

f p (x)

L 1 ((p − 1)x) L 1 (p − 1) − x

x ^−α−3 dx, L 1 (x) = xL(x + 1), δ 2 (p) = 1

I 2

Z +∞

1 g p (x)

L 2 (px) L 2 (p) − 1

x

x ^−α−1 dx, L 2 (x) = L(x)/x.

Since

Z p+1 p

η(t) t dt = O

|η(p)|

p

,oneclearlyhas

µ p

µ p+1

− M p

M p+1 = 1 θ

1 − 1

p + 1 −α

R M (p) + R δ (p)

R M (p + 1) + R δ (p + 1) − R M (p) R M (p + 1)

+ O

|η(p)|

p

,

⁽⁵⁾

anditisstraightforwardthat

R M (p) + R δ (p)

R M (p + 1) + R δ (p + 1) − R M (p) R M (p + 1)

= R δ (p) R M (p + 1) − R δ (p + 1) R M (p) [R M (p + 1) + R δ (p + 1)] R M (p + 1)

= [R δ (p) − R δ (p + 1)] R M (p + 1) − R δ (p + 1)[R M (p) − R M (p + 1)]

[R M (p + 1) + R δ (p + 1)] R M (p + 1) .

Lemma7entailsthat

R M → 1

^and

R δ → 0

^as

p → ∞

^and

R δ (p + 1) = O (|η(p)| (1 + L(p))) , R M (p) − R M (p + 1) = O 1/p ²

, R δ (p) − R δ (p + 1) = O (|η(p)|/p) ,

where

L

îs^slowly^varyingâtînnity. Consequently,

R M (p) + R δ (p)

R M (p + 1) + R δ (p + 1) − R M (p) R M (p + 1) = O

|η(p)|

p + |η(p)| (1 + L(p)) p ²

= O

|η(p)|

p

,

andreplacingin (5)yieldsthedesiredresult.

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ApplyingProposition3enablesustocontrolthebiastermintroducedwhen

M p n

^is^replaced^by

µ p n

^:

1 Θ n

= 1 θ + O

|η(p n )|

p n

.

⁽⁶⁾

Wenowturnto therandomterm:

Theorem2. Assumethat

(A 1 )

^and

(A 2 )

^hold. ^If

n p ^−α _n L(p n ) → ∞

^then

v n

b θ n

Θ n

− 1

!

−→ N d (0, V (α, a))

^as

n → ∞,

with

v n = p

n L(p n ) p ^−α/2+1 n

^and

V (α, a) = α + 1 a ² Γ(α)

2 ^−α−2 − 2 (a + 1) ^α+1

(a + 2) ^α+2 + 2 ^−α−2 (a + 1) ^α

.

Proof. Ourgoalistoprovethatthesequenceofrandomvariables

(ξ n )

^dened ^by

ξ n = θ

p V (α, a) v n

1 b θ n

− 1 Θ n

converges in distribution to a standard Gaussian random variable, Theorem 2 then being a

simpleconsequenceofthisresult.

Therststepconsistsin usingLemma9inordertolinearize

ξ n

^:

ξ n = u n, a

ζ _n ⁽¹⁾ +

µ p n +1

b µ p n +1

− 1

ζ _n ⁽²⁾ +

1 + ap n

p n + 1

µ (a+1)p n +1

b

µ (a+1)p n +1

− 1

ζ _n ⁽³⁾

(1 + o(1))

= u n, a

h ζ _n ⁽¹⁾ + o ^P (ζ _n ⁽²⁾ ) + o ^P (ζ _n ⁽³⁾ ) i

(1 + o(1)),

inviewofProposition2. Thus,toconcludetheproof,itisenoughtoshowthat

u n, a ζ _n ⁽¹⁾ −→ N ^d (0, 1),

^(7a)

u n, a ζ _n ⁽²⁾ −→ N ^d (0, C 2 ),

^(7b)

u n, a ζ _n ⁽³⁾ −→ N ^d (0, C 3 ),

^(7c)

where

C 2

^and

C 3

^are^suitable^constants. ^Let^us^then^write

ζ n ⁽¹⁾ = P n

k=1 S _{n, k} ⁽¹⁾

^,^where

S _{n, k} ⁽¹⁾ = 1

n A ^t _n h

X _k ^p ⁿ , X _k ^p ⁿ ⁺¹ , X _k ^(a+1)p ⁿ , X _k ^(a+1)p ⁿ ⁺¹ i t

, A n = h

a ⁽¹⁾ _n, ₀ , a ⁽¹⁾ _n, ₁ , a ⁽¹⁾ _n, ₂ , a ⁽¹⁾ _n, ₃ i t

, a ⁽¹⁾ _n, ₀ = −1,

a ⁽¹⁾ _n, ₁ = µ p n

µ p n +1

, a ⁽¹⁾ _n, ₂ =

1 + ap n

p n + 1

µ p n +1

µ (a+1)p n +1

, a ⁽¹⁾ _n, ₃ = −

1 + ap n

p n + 1

µ p n +1 µ (a+1)p n

µ ² _(a+1)p _n ₊₁ ,

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with

A ^t

^standing ^for ^the ^transposed ^matrix ^of

A

^. În ôrder ^to ûse Lyapounov's central limit theorem(seee.g. Billingsley,1979,p. 312),itremainstoprovethat

1 [Var(ζ n ⁽¹⁾ )] ^3/2

X n k=1

E |S _{n, k} ⁽¹⁾ | ³ → 0

⁽⁸⁾

as

n → ∞

^,^which ^requires^to^control

Var(ζ n ⁽¹⁾ )

^and

E |S _n, ⁽¹⁾ ₁ | ³

^.

Tocomputeanequivalentfor

Var(ζ n ⁽¹⁾ )

^,^remark^that

Var(ζ n ⁽¹⁾ ) = 1

n A ^t _n M n A n

^with

M n =



 



µ 2p n µ 2p n +1 µ (a+2)p n µ (a+2)p n +1

µ 2p n +1 µ 2p n +2 µ (a+2)p n +1 µ (a+2)p n +2

µ (a+2)p n µ (a+2)p n +1 µ (2a+2)p n µ (2a+2)p n +1

µ (a+2)p n +1 µ (a+2)p n +2 µ (2a+2)p n +1 µ (2a+2)p n +2



 

 .

Letusnowrewritethatas

Var(ζ _n ⁽¹⁾ ) = 1 n

w(p n , p n ) − 2

1 + ap n

p n + 1

µ p n +1

µ (a+1)p n +1

w(p n , (a + 1)p n ) +

1 + ap n

p n + 1

2 µ ² _p _n ₊₁

µ ² _(a+1)p _n ₊₁ w((a + 1)p n , (a + 1)p n )

#

where

w(up n , vp n ) =

−1, µ up n

µ up n +1



 µ (u+v)p n µ (u+v)p n +1

µ (u+v)p n +1 µ (u+v)p n +2





−1, µ vp n

µ vp n +1

t

.

WenowapplyProposition3,anduse(4)togetherwithLemma 7toobtain,aftersomecumber-

somebut elementarycomputations,

w(up n , vp n ) = Γ(α + 1) α(α + 1)

(u + v) ^α+2 θ ^(u+v)p ⁿ p ^−α−2 _n L(p n ) (1 + o(1)).

Takingintoaccountthat

1 + ap n

p n + 1

µ p n +1

µ (a+1)p n +1

= (a + 1) ^α+1

θ ^ap ⁿ (1 + o(1))

⁽⁹⁾

weget

Var(ζ _n ⁽¹⁾ ) = a ² Γ ² (α + 1)V (α, a) 1

n θ ^2p ⁿ p ^−α−2 _n L(p n ) (1 + o(1)).

⁽¹⁰⁾

Toshow(8),itthensucestoprovethat

E |S _n, ⁽¹⁾ ₁ | ³ = O(n ⁻³ θ ^3p ⁿ p ^−α−3 _n L(p n )).

^To^this^aim,^let

us introduce

Y 1 = X 1 /θ

^and ^the ^associated ^survival ^function

F 1 (x) = (1 − x) ^α L((1 − x) ⁻¹ )

^,

∀ x ∈ [0, 1]

^. ^Hölder's^inequality^leads^to

E |S _n, ⁽¹⁾ ₁ | ³

n ⁻³ θ ^3p ⁿ ≤ 4 E |Y ₁ ^p ⁿ (a ⁽¹⁾ _n, ₀ + a ⁽¹⁾ _n, ₁ θ Y 1 )| ³ + 4 E |Y ₁ ^(a+1)p ⁿ (a ⁽¹⁾ _n, ₂ θ ^ap ⁿ + a ⁽¹⁾ _n, ₃ θ ^ap ⁿ ⁺¹ Y 1 )| ³ .

(9)

H _n, ⁽¹⁾ ₀ (u) = −1, H _n, ⁽¹⁾ ₁ (u) = αu, H _n, ⁽¹⁾ ₂ (u) =

1 + ap n

p n + 1

θ ^ap ⁿ µ p n +1

µ (a+1)p n +1

, H _n, ⁽¹⁾ ₃ (u) = −

1 + ap n

p n + 1

θ ^ap ⁿ µ p n +1

µ (a+1)p n +1

· αu a + 1 ,

somemorestraightforwardalbeitburdensomecomputationsshowthatthereexisttwosequences

of Borel functions

(χ ^(1,1) n )

^and

(χ ^(1, n ²⁾ )

^uniformly ^converging ^to

0

^on

[0, 1]

^such ^that ^for ^all

u ∈ [0, 1]

^,

a ⁽¹⁾ _n,0 + a ⁽¹⁾ _n, ₁ θu = H _n, ⁽¹⁾ ₀ (u)(1 − u) + H _n, ⁽¹⁾ ₁ (u) + χ ^(1, n ¹⁾ (u)

p n ,

a ⁽¹⁾ _n, ₂ θ ^ap ⁿ + a ⁽¹⁾ _n, ₃ θ ^ap ⁿ ⁺¹ u = H _n, ⁽¹⁾ ₂ (u)(1 − u) + H _n, ⁽¹⁾ ₃ (u) + χ ^(1, n ²⁾ (u) p n

.

Recalling(9), weobtainthat

H _{n, j} ⁽¹⁾

âre^Borelûniformly^bounded ^functionsôn

[0, 1]

^,^so^that ^we

canuseLemma10twicetoobtainthedesiredboundfor

E |S _n, ⁽¹⁾ ₁ | ³

^. ^Finally,^applyingLyapounov's central limittheoremandusing thecondition

n p ^−α _n L(p n ) → ∞

^concludes ^the^proof^of^(7a).

Proofsof(7b)and(7c)arecompletelysimilarsince

ζ n ⁽²⁾

^and

ζ n ⁽³⁾

^can^be^rewritten^as

ζ _n ⁽²⁾ = X n k=1

S _{n, k} ⁽²⁾

^with

S _{n, k} ⁽²⁾ = 1 n

h a ⁽²⁾ _n, ₀ , a ⁽²⁾ _n, ₁ i h

X _k ^p ⁿ , X _k ^p ⁿ ⁺¹ i t

,

ζ _n ⁽³⁾ = X n k=1

S _{n, k} ⁽³⁾

^with

S _{n, k} ⁽³⁾ = 1 n

h

a ⁽³⁾ _n, ₀ , a ⁽³⁾ _n, ₁ i h

X _k ^(a+1)p ⁿ , X _k ^(a+1)p ⁿ ⁺¹ i t

withcleardenitionsofthesequences

a ^(j) _{n, i}

^,

i = 0, 1

^,

j = 2, 3

^. ^Applying^Lemma¹⁰^with

H _n, ⁽²⁾ ₀ (u) = −1,

H _n, ⁽²⁾ ₁ (u) = αu,

H _n, ⁽³⁾ ₀ (u) = θ ^ap ⁿ µ p n +1

µ (a+1)p n +1

, H _n, ⁽³⁾ ₁ (u) = −θ ^ap ⁿ µ p n +1

µ (a+1)p n +1

· αu a + 1

yields

E |S _n, ^(j) ₁ | ³ = O(n ⁻³ θ ^3p ⁿ p ^−α−3 _n L(p n ))

^,

j = 2, 3

^. ^UsingLyapounov'scentral limittheorem thenallowsustocompletetheproofofTheorem2.

Noticingthat

θ b n − θ = Θ n

"

θ b n

Θ n

− 1

#

+ [Θ n − θ]

^,^the^asymptotic^normality^of

θ b n

^centered^on^the

trueendpoint

θ

^is^aconsequenceof(6)andTheorem2.

Theorem3. Assumethat

(A 1 )

^and

(A 2 )

^hold. ^If

n p ^−α _n L(p n ) → ∞

^and

n p ^−α _n L(p n )η ² (p n ) → 0

^,

then

v n

b θ n

θ − 1

!

−→ N d (0, V (α, a))

^as

n → ∞,

(10)

Inviewof Theorem3,it maybe interestingtoestimate theunknownparameter

α

^. ^F^rom ^(3),

thefollowingestimatorisconsidered:

b

α n = (p n + 1)

θ b n µ b p n

b µ p n +1

− 1

.

Proposition4. Under theassumptions of Theorem 3,

α b n = α + O ^P (p n /v n )

^.

Proof. Letusintroduce

α n = (p n + 1)

Θ n

µ p n

µ p n +1

− 1

andfocusrstontherandomterm

v n

p n

( α b n − α n ) = v n

" h

θ b n − Θ n i b µ p n

b µ p n +1

− Θ n

µ p n +1

b µ p n +1

· ζ n ⁽²⁾

µ p n +1

#

(1 + o(1))

withnotationsof Lemma9. Recallthat, from Proposition1,

µ p n /µ p n +1 → 1/θ

^, ^from^Proposi-

tion2,

µ p n / µ b p n

−→ P 1

^and^from ^(6),

Θ n → θ

^as

n → ∞

^so^that

v n

p n ( α b n − α n ) = v n

θ b n − Θ n

1 θ + o ^P (1)

− θv n

ζ n ⁽²⁾

µ p n +1 (1 + o ^P (1)).

Besides,applyingTheorem2yields

v n

θ b n − Θ n

= O P (1)

^. ^Now,

v n

ζ n ⁽²⁾

µ p n +1

= v n

µ p n +1 u n, a

u n, a ζ _n ⁽²⁾ = O ^P (1),

from Lemma 8 and since

u n, a ζ n ⁽²⁾

^is asymptotically Gaussian (see (7b)). As a preliminary conclusion,wehave

v n

p n

( α b n − α n ) = O ^P (1).

Turningtothebiasterm,(6)andProposition3yield

α n = α + (p n + 1) O

|η(p n )|

p n

= α + o p n

v n

,

whichcompletestheproof.

Byplugging

α b n

ⁱⁿ^theâsymptotic^varianceôf^Theorem ^3,^classicalârguments^thus^yield:

Corollary 1. Underthe assumptions ofTheorem 3,

v n

s 1 V ( α b n , a)

θ b n

θ − 1

!

−→ N(0, d 1)

^as

n → ∞.

Condence intervalsfor

θ

^may^then^be^built^using^this^result.

5 Examples

Inthissection, wehighlightsomecaseswhereourhypotheseshold. Since

η(x) = xL ⁰ (x)/L(x)

^,

onecanseethat

(A 1 )

^and

(A 2 )

^are^satisedⁱⁿ^the^general^context^of:

(11)

1. TheHallmodel(seeHall,1982),namely

L(x) = C+Dx ^−β (1+δ(x))

^for^all^suciently^large

x

^,^where

C, β > 0

^,

D ∈ R \ {0}

^and

δ

^is^a^Borel^bounded ^twicecontinuouslydierentiable function on

(1, +∞)

^such^that

δ(x) → 0

^,

xδ ⁰ (x) → 0

^and

x ² δ ⁰⁰ (x) → 0

^as

x → +∞

^. ^Here,

ν = −β < 0

^.

2. Thecasewhere

L(x) = f (ln x)

^,^where

f

^is^a^rational^function. ^Here,

ν = 0

^.

Letusnowfocusontwoparticulardistributionsthatarealsousedforthenumericalexperiments

ofSection 6. Bothofthemhaveanendpoint

θ = 1

^. ^The^rstdistributionhassurvivalfunction

F (x) =

"

1 + 1

x − 1

−τ ₁ # −τ ₂

, x ∈ (0, 1),

⁽¹¹⁾

with

τ 1 , τ 2 > 0

^. ^Remark ^that, ^if

X

^is distributed from (11), then it can be rewritten as

X = 1 − 1/(1 + Y )

^where

Y

^is ^Burr(

1, τ 1 , τ 2

⁾ distributed, namely,

Y

^has ^survival ^function

G(y) = (1 + y ^τ ¹ ) ^−τ ²

^for

y ≥ 0

^. ^It^can^be^shown^that

(A 1 )

^is^veried^with

α = τ 1 τ 2

^and

∀ y ≥ 1, L(y) =

y ^τ ¹ 1 + (y − 1) ^τ ¹

τ ₂

. L

^is^clearly

C ^∞

^on

(1, +∞)

ândône^readilyôbtains

∀ y > 1, η(y) := y L ⁰ (y)

L(y) = τ 1 τ 2 1 − (y − 1) ^τ ¹ ⁻¹ 1 + (y − 1) ^τ ¹ .

As aresult,

η

^iscontinuously dierentiableon

(1, +∞)

^, ûltimately^monotonic ând^non îdenti-

cally

0

^. ^Besides,

y η ⁰ (y)

η(y) = −y

(τ 1 − 1) (y − 1) ^τ ¹ ⁻² 1 − (y − 1) ^τ ¹ ⁻¹ + τ 1

(y − 1) ^τ ¹ ⁻¹ 1 + (y − 1) ^τ ¹

→ − min(τ 1 , 1) < 0,

as

y → +∞

^and ^thus

(A 2 )

^holds ^with

ν = − min(τ 1 , 1)

^. ^Note ^that ^one^can^also ^show ^that

L

belongstotheHallclass. Thesecond considereddistributionhassurvivalfunction

F (x) = 1 Γ(b)

Z ∞

− ln(1−x)

(λt) ^b−1 λe ^−λt dt, x ∈ (0, 1),

⁽¹²⁾

with

b ≥ 1

^and

λ > 0

^. ^Here,^when

X

^isdistributedfrom(12),itcanberewrittenas

X = 1 −e ^−Y

where

Y

^is^Gamma(

b, λ

⁾distributed. Notethat,if

b = 1

^,^then

X

^has^survival^function

F (x) = (1 − x) ^λ

^, ^namely

L ≡ 1

^,^and

(A 1 )

^,

(A 2 )

straightforwardlyhold. If

b > 1

^,^then

(A 1 )

^holds^with

α = λ

^,

L(x) = λ ^b−1

Γ(b) ln ^b−1 (x)[1 + δ(x)]

δ(x) = 1

x ^−λ λ ^b−1 ln ^b−1 (x) Z ∞

ln x

(λt) ^b−1 λe ^−λt dt

− 1 = (b − 1) Z ∞

1 u ^b−2 e ^{−λ(u−1) ln} ^x du.

(12)

Notethat

δ

^is

C ^∞

^on

(1, +∞)

^and^goes^to

0

^at ^innity. ^Therefore,

L

^is^slowly^varying ^and

C ^∞

on

(1, +∞)

^. ^Now

η(x) := x L ⁰ (x)

L(x) = b − 1

ln x + xδ ⁰ (x)(1 + o(1))

= b − 1

ln x − λ(b − 1) Z ∞

1 (u − 1)u ^b−2 e ^{−λ(u−1) ln} ^x du(1 + o(1))

= b − 1

ln x + o(1/ ln x),

sothat

η

îs^slowly^varyingând^positiveⁱⁿ âneighborhoodof

+∞

^. ^Finally,^noting^that

d

dx

h xδ ⁰ (x) i

= λ ² (b − 1) x

Z ∞ 1

(u − 1) ² u ^b−2 e ^{−λ(u−1) ln} ^x du = o 1

x ln ² x

it followsthat

η ⁰ (x) = (1 − b)

x ln ² x (1 + o(1))

^entailing^that

η

^is^ultimately non-increasingand that

x η ⁰ (x)/η(x) → 0

^as

x → +∞

^. ^As^aconclusion,

(A 2 )

^holds^with

ν = 0

^.

6 Numerical experiments

Inthissection,weshallexaminetheperformancesofourestimatoronsampleswithsize

n = 500

oneightsituationsobtainedbyconsideringthemodels(11)and(12)withfourdierentsetsof

parameters, see the rst column of Table 1. We choose

p n = n ^1/α / ln ln n

ⁱⁿ ^order ^to ^satisfy

the assumptions in Theorem 3and aset

A = {0.2, 0.6, 1.0, . . . , 21}

^of ^dierent ^values^of

a

^is

tested. Ineach ofthe eightsituations,

N = 1000

replications ofthesample aregenerated and theaverage

L ¹ −

^error^is^computed:

E(a) = 1 N

X N j=1

|ε(j, a)| ,

^where

ε(j, a) = θ b ^{(j, a)} − θ

with

θ b ^{(j, a)}

^being^the^estimator^computed^on^the

j−

^threplicationwith

a ∈ A

^and

θ = 1

^. ^Then,

theoptimal valueof

a

^is^retained:

a ^? = argmin{E(a), a ∈ A}.

^F^or^the^sake^ofcomparison,the same procedure hasbeen applied to the extreme-valuemoment estimator, see for instance de

HaanandFerreira(2006,Remark4.5.5),whichdepends onaparameter

k ∈ {2, 3, . . . , n − 1}

^.

Thenaivemaximumestimatorhasalsobeenconsidered. Notethat,sincethemaximumestima-

tordoesnotdependonanyparameter,theassociatedfunction

E

^is^constant. ^Numerical^results

aresummarizedinTable1,where

E(a ^? )

îs^displayed. În^theûpper^partôf^the^table,îtâppears

that, for the distribution (11), performance of all these estimators decrease as

|ν|

^decreases.

This phenomenon canbe explained since

ν

^drives ^the ^bias ^of ^most extreme-valueestimators.

Forinstance, when

|ν |

^is ^small,

η

^converges^slowly^to ⁰^and Proposition 3 showsthat theap- proximationerrorof

µ p /µ p+1

^by

M p /M p+1

îs ^large. ^Besides, ^the ^lower^partôf ^Tâble ¹^shows

that, forthe distribution(12), when

α

^increases, performance of all these estimatorsdecrease aswell, sincethe simulatedpoints are gettingmoreand moredistantfrom the endpoint. Let

(13)

resultsthanthemaximumestimatorandtheextreme-valuemomentestimator.

To further compare the behavior of the estimators in the optimal case, boxplots of the

associatederrors

ε(j, a ^∗ )

âre^displayedôn^Figure¹ând^Figure^2. ^Clearly,^the^maximumâs^well

asourestimatorunderestimatetheendpoint. However,theerrorassociatedtoourestimatoris

smallerthantheerrorofthemaximum. Besides,ourestimatorhasasmallervariancethanboth

themaximumestimatorandtheextreme-valuemomentestimator.

Agraphicalcomparisononbothmodelsofthefunctions

E

^associated^to^the^three^estimators

isproposedonFigure36. Onmodel(12),theshapeofthecurvesassociatedto ourestimator

and to the extreme-value moment estimator are similar, see Figure 5 and Figure 6. On the

contrary,itappearsonFigure3andFigure4that,onmodel(11),thefunctions

E

^associated^to

theextreme-valuemomentestimatorandourestimatorhaveverydierentshapes,eventhough

theyhavesimilarminima. Theerrorassociatedto theextreme-valuemomentestimatorisvery

sensitivetothechoiceoftheparameter

k

^whereas^theêrrorâssociated^toôurêstimatorîs^stable

foralargepanelof

a

^values.

References

Athreya,K.B., Fukuchi,J. (1997). Condence intervalsfor endpoints ofac.d.f viabootstrap.

J. Statist. Plann. Inference58:299320.

Billingsley,P. (1979). Probability andmeasure,JohnWileyandSons.

Bingham,N.H.,Goldie,C.M.,Teugels,J.L.(1987). RegularVariation,Cambridge,U.K.: Cam-

bridgeUniversityPress.

Chow,Y.S.,Teicher,H.(1997). Probability Theory,Springer.

Cooke,P. (1979). Statisticalinferenceforbounds ofrandomvariables. Biometrika66:367374.

Dekkers,A.L.M.,Einmahl,J.H.J.,deHaan,L.(1989). Amomentestimatorfortheindex ofan

extreme-valuedistribution. Ann. Statist. 17:18331855.

Drees,H., Ferreira,A., deHaan,L.(2003). Onmaximumlikelihoodestimationoftheextreme

valueindex. Ann. Appl. Probab. 14:11791201.

Goldenshluger,A.,Tsybakov,A.(2004). Estimatingtheendpointofadistributionin thepres-

enceofadditiveobservationerrors. Statist. Probab. Lett. 68:3949.

de Haan,L.(1981). Estimationof theminimumof afunctionusing orderstatistics. J. Amer.

Statist. Assoc. 76:467469.

deHaan,L.,Ferreira,A. (2006). ExtremeValue Theory,Springer.

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Hall, P., Wang, J.Z. (1999). Estimating the end-point of a probability distribution using

minimum-distancemethods. Bernoulli 5(1):177189.

Hall, P., Wang, J.Z. (2005). Bayesian likelihood methods for estimating the end point of a

distribution. J. Roy. Statist. Soc. Ser. B 67(5):717729.

Hosking, J.R.M., Wallis, J.R. (1987). Parameterand quantile estimation for the generalized

Paretodistribution. Technometrics29:339349.

Li,D.,Peng,L.(2009). Doesbiasreductionwithexternalestimatorofsecondorder parameter

workforendpoint? J.Statist. Plann. Inference139:19371952.

Loh, W.Y. (1984). Estimating an endpoint of adistribution with resampling methods. Ann.

Statist. 12(4):15431550.

Miller,R.G.(1964). Atrustworthyjackknife. Ann. Math. Statist. 35:15941605.

Neves,C.,Pereira,A.(2010). Detectingnitenessintherightendpointoflight-taileddistribu-

tions. Statist. Probab. Lett. 80:437444.

Quenouille,M.H.(1949). Approximatetestsofcorrelationin times-series. J. Roy. Statist. Soc.

Ser. B 11:6884.

Robson,D.S.,Whitlock,J.H.(1964). Estimationofatruncationpoint. Biometrika51:3339.

Smith, R.L. (1987). Estimating tailsof probability distributions. Ann. Statist. 15(3):1174

1207.

Smith,R.L.,Weissman, I.(1985). Maximumlikelihood estimationofthelowertailofaproba-

bilitydistribution. J. Roy. Statist. Soc. Ser. B 47:285298.

(15)

Letusset

F 1 (y) := F(θy)

^and

µ 1, p n := µ p n /θ ^p ⁿ

^. ^The^rst^result^deals^with^the^behavior^of^the

moment

µ 1, p n

^.

Lemma1. If

(A 0 )

^holds,^then

µ 1, p n /µ 1, p n +1 → 1

^as

n → ∞

^.

As it has been mentionedbefore,

(A 2 )

^implies ^that

x η ⁰ (x) → 0

^as

x → ∞

^. ^The^next ^lemma

establishessomeconsequencesofthisproperty.

Lemma 2. Let

ϕ

^be ^acontinuously dierentiable function on

(1, +∞)

^such^that

x ϕ ⁰ (x) → 0

as

x → +∞

^. ^Then,

(i)

t sup

x≥1

|ϕ(tx) − ϕ((t + 1)x)| → 0

^as

t → ∞

^.

(ii)Forall

q > 0

^,

t sup

x∈(0,1]

x ^q |ϕ(tx) − ϕ((t + 1)x)| → 0

^as

t → ∞

^.

Before proceeding, letus introduce somemore notations. For all

k ∈ R

, let

P k

^be ^the ^set ^of

collectionsof Borelfunctions

(f p ) p≥1

^on

(0, 1]

^such^that

1.

∃ p k ≥ 1, ∃ C k ≥ 0, ∀ p ≥ p k , ∀ x ∈ (0, 1], |f p (x)| ≤ C k x ^k

^,

2.

∃ p k ≥ 1, ∃ C k ≥ 0, ∀ p ≥ p k , ∀ x ∈ (0, 1], p ² |f p+1 − f p |(x) ≤ C k x ^k

^,

3.

∀ x ∈ (0, 1], p ² |f p+2 − 2f p+1 + f p |(x) → 0

^as

p → ∞

^.

Let

P = \

k≥0

P k

^. ^Besides, ^let

U

^be ^the^set ^ofcollectionsofBorelfunctions

(f p ) p≥1

^on

[1, +∞)

suchthat

1.

sup

x≥1

|f p (x)| = O(1)

^as

p → ∞

^,

2.

p ² sup

x≥1

|f p+1 − f p |(x) = O(1)

^as

p → ∞

^,

3.

p ² sup

x≥1

|f p+2 − 2f p+1 + f p |(x) → 0

^as

p → ∞

^.

These sets will reveal useful to study the asymptotic properties of

θ b n

^since ^this ^estimator ^is

basedonincrementsofsequencesoffunctions. Astabilitypropertyof theset

P

^is ^givenⁱⁿ^the

nextlemma.

Lemma 3. Let

(f p )

^,

(g p )

^be^two collections of Borelfunctions. If for some

k ∈ R

,

(f p ) ∈ P k

and

(g p ) ∈ P

^,^then

(f p g p ) ∈ P

^.

Wenowgiveacontinuitypropertyofsomeintegraltransformsdened on

P

^and

U

^.

Lemma4. Let

(f p ) ∈ P

^,

(g p ) ∈ U

^and

(u p )

^,

(v p )

^be^twocollectionsofBorelfunctionssuchthat

f p (x) → f (x)

^for ^all

x ∈ (0, 1]

^,

sup

x≥1

|g p (x) − g(x)| → 0, sup

0<x≤1

|u p (x) − u(x)| → 0

^and

sup

x≥1

|v p (x) − v(x)| → 0

^as

p → ∞,

(16)

where

f, g, u, v

^are^four^Borel^functions^such^that

f

^and

u

^(resp.

g

^and

v

⁾ ^are^dened^on

(0, 1]

(resp.

[1, +∞)

^). ^Assume^further^that

u

^and

v

^are^bounded. ^Then, ^for ^all

k > 1

^,

Z 1

0 x ^−k f p (x) u p (x) dx → Z 1

0 x ^−k f (x) u(x) dx, Z +∞

1 x ^−k g p (x) v p (x) dx →

Z +∞

1 x ^−k g(x) v(x) dx

as

p → ∞

^.

Thefollowinglemma providessucient conditionsoncollectionsoffunctions to belong tothe

previoussets.

Lemma 5. Let

(f p )

^,

(g p )

^be ^twocollections of Borel functions. Assume that thereexist Borel functions

F i

^and^Borel^bounded^functions

G i

^,

0 ≤ i ≤ 2

^,^such^that

∀ x ∈ (0, 1], p ² f p (x) −

X 2 k=0

p ^−k F k (x)

→ 0

^as

p → ∞, p ² sup

x≥1

g p (x) −

X 2 k=0

p ^−k G k (x)

→ 0

^as

p → ∞.

Then, for all

x ∈ (0, 1]

^,

p ² |f p+2 − 2f p+1 + f p |(x) → 0

^as

p → ∞

^,^and

(g p ) ∈ U

^.

Wearenowinpositionto exhibittwoparticularelementsof

P

^and

U

^:

Lemma6. Let

(f p )

^and

(g p )

^,

p ≥ 1

^be^twocollectionsof Borelfunctionsdenedby

∀ x ∈ (0, 1], f p (x) =

1 − 1 p

−α−1

1 + 1

(p − 1)x

−α−2

1 − 1

(p − 1)x + 1 p−1

,

∀ x ∈ [1, +∞), g p (x) =

1 − 1 px

p−1

.

Then

(f p ) ∈ P

^,

(g p ) ∈ U

^and

∀ x ∈ (0, 1], f p (x) → e ^−1/x

^and

sup

x≥1

g p (x) − e ^−1/x → 0

^as

p → ∞.

⁽¹³⁾

Lemma 7isthekeytoolforestablishingpreciseexpansionsofthemoments

µ p

^and

M p

^.

Lemma7. Let

(f p ) ∈ P

^and

(g p ) ∈ U

^such^that ⁽¹³⁾^holds^and^dene

E 1 (p) = 1

I 1

Z 1 0

f p (x)x ^−α−2 dx − 1, I 1 = Z +∞

1 y ^α e ^−y dy, E 2 (p) = 1

I 2

Z +∞

1 g p (x)x ^−α−2 dx − 1, I 2 = Z 1

0 y ^α e ^−y dy, δ 1 (p) = 1

I 1

Z 1 0

f p (x)

L 1 ((p − 1)x) L 1 (p − 1) − x

x ^−α−3 dx, L 1 (x) = xL(x + 1), δ 2 (p) = 1

I 2

Z +∞

1 g p (x)

L 2 (px) L 2 (p) − 1

x

x ^−α−1 dx, L 2 (x) = L(x)/x,

where

L

îsâ^slowly^varying ^function âtînnity. ^Then,^for âll

i = 1, 2

^,

(17)

(i)

E i (p) → 0

^as

p → ∞

^,

(ii)

p ² (E i (p + 1) − E i (p)) = O(1)

^,

(iii)

p ² (E i (p + 2) − 2 E i (p + 1) + E i (p)) → 0

^as

p → ∞

^,

(iv)

δ i (p) → 0

^as

p → ∞

^.

Moreover,if

L

^satises

(A 2 )

^,^then

(v)Thereexistsaslowly varyingfunction

L

^such^that

δ 1 (p) = O (|η(p)| L(p))

^,

(vi)

δ 2 (p) = O (|η(p)|)

^,

(vii)Forall

i = 1, 2

^,

δ i (p + 1) − δ i (p) = O (|η(p)|/p)

^,

(viii)For all

i = 1, 2

^,

p ² (δ i (p + 2) − 2 δ i (p + 1) + δ i (p)) → 0

^as

p → ∞

^.

Sometimes,arstorderexpansionofthemoment

µ p

^is^sucient:

Lemma8. If

(A 1 )

^holds^then,^as

p → ∞

^,

µ p = p ^−α θ ^p L(p) Γ(α + 1)(1 + o(1)).

Thenextresultconsistsin linearizingthequantity

ξ n

^appearingⁱⁿ ^the^proof^of ^Theorem^2:

Lemma9. Let

p n → ∞

^and

ν p = µ b p − µ p

^. ^If

(A 1 )

^is^satised, ^then

ξ n = u n, a

ζ _n ⁽¹⁾ +

µ p n +1

b µ p n +1

− 1

ζ _n ⁽²⁾ +

1 + ap n

p n + 1

µ (a+1)p n +1

b

µ (a+1)p n +1

− 1

ζ _n ⁽³⁾

(1 + o(1)),

where

ζ _n ⁽¹⁾ = ζ _n ⁽²⁾ +

1 + ap n

p n + 1

ζ _n ⁽³⁾ ,

with

ζ _n ⁽²⁾ = −ν p n + µ p n

µ p n +1 ν p n +1 , ζ _n ⁽³⁾ = µ p n +1

µ (a+1)p n +1

ν (a+1)p n − µ (a+1)p n

µ (a+1)p n +1

ν (a+1)p n +1

and

u n, a = 1 a Γ(α + 1)

s 1 V (α, a)

p ^α _n v n

θ ^p ⁿ L(p n ) .

Thenal lemma ofthis sectionprovidesanasymptotic bound of thethird-ordermomentsap-

pearingintheproofofTheorem2.

Lemma10. Let

k ∈ N

and

p n → ∞

^. ^Let

(H n, j ) 0≤j≤k

^be^sequences^of^Borel^uniformly^bounde^d

functionson

[0, 1]

^and

∀ u ∈ [0, 1], h n (u) = X k j=0

H n, j (u) p ^j n

(1 − u) ^k−j .

If

Y

^is^a^random ^variable ^with^survival^function

G(x) = (1 − x) ^α L((1 − x) ⁻¹ )

^where

α > 0

^and

L

îsâ^Borel^slow^ly^varying^function âtînnity, ^then

E |Y ^p ⁿ h n (Y )| ³ = O(p ^−α−3k _n L(p n )).

(18)

Proof ofLemma1. Let

I p n := µ 1, p n /p n

^and

ε > 0

^. ^The^integral

I p n

îsêxpandedâs

I p n = Z 1

1−ε

y ^p ⁿ ⁻¹ F 1 (y) dy



 

  1 + Z 1−ε

0 y ^p ⁿ ⁻¹ F 1 (y) dy Z 1

1−ε

y ^p ⁿ ⁻¹ F 1 (y) dy



 

 

where

0 ≤ Z 1−ε

0 y ^p ⁿ ⁻¹ F 1 (y) dy Z 1

1−ε

y ^p ⁿ ⁻¹ F 1 (y) dy

≤ 1 − ε

Z 1 1−ε

y 1 − ε

p n −1

F 1 (y) dy

≤ 1 − ε

1 − ε/2 1 − ε

p n −1 Z 1 1−ε/2

F 1 (y) dy .

Since

1 − ε/2 1 − ε

p n −1

→ ∞

^as

n → ∞

^,^it^follows^that

I p n = Z 1

1−ε

y ^p ⁿ ⁻¹ F 1 (y) dy (1 + o(1)).

⁽¹⁴⁾

Inviewof

1 ≤ Z 1

1−ε

y ^p ⁿ ⁻¹ F 1 (y) dy Z 1

1−ε

y ^p ⁿ F 1 (y) dy ≤ 1 1 − ε

and(14),onethushas

I p n /I p n +1 → 1

^as

n → ∞

^and^Lemma¹^is^proved.

Proof of Lemma 2. If

ϕ ⁰

^isidentically

0

^, ^then

ϕ

^is^constant^on

[1, +∞)

^and ^the^results^are

straightforward. Otherwise,letusconsider(i)and(ii)separately.

(i)Let

t, x ≥ 1

^. ^The^mean^value^theorem^shows^that^there ^exists

h 1 (t, x) ∈ (0, 1)

^such ^that

t |ϕ(tx) − ϕ((t + 1)x)| = t

t + h 1 (t, x) |[(t + h 1 (t, x))x] ϕ ⁰ [(t + h 1 (t, x))x]|

≤ sup

y≥t

|y ϕ ⁰ (y)| → 0

uniformlyin

x ≥ 1

^,^as

t → +∞

^.

(ii)Let

t ≥ 1

^and

x ∈ (0, 1]

^,

q > 0

^,

ε > 0

^and

c(ε) :=

ε

2 · 1

sup _y>1 |y ϕ ⁰ (y)|

1/q

.

Applyingthemeanvaluetheoremagainshowsthatthere exists

h 2 (t, x) ∈ (0, 1)

^such ^that

tx ^q |ϕ(tx) − ϕ((t + 1)x)| = x ^q t

t + h 2 (t, x) |[(t + h 2 (t, x))x] ϕ ⁰ [(t + h 2 (t, x))x]|

≤ x ^q sup

y>1

|y ϕ ⁰ (y)|1l {0<x<c(ε)} + sup

y≥t c(ε)

|y ϕ ⁰ (y)|1l {c(ε)≤x≤1}

≤ ε 2 + ε

2 = ε

forall

t

^large^enough,^uniformlyⁱⁿ

x ∈ (0, 1]

^,^which ^concludes^the^proof^of^Lemma^2.

(19)

(f g) p+1 − (f g) p = f p+1 (g p+1 − g p ) + g p (f p+1 − f p ),

(f g) p+2 − 2(f g) p+1 + (f g) p = (f p+2 − 2f p+1 + f p ) g p+2 + (f p+1 − f p ) (g p+2 − g p ) + f p+1 (g p+2 − 2g p+1 + g p ),

andfromthepropertiesof

(f p )

^and

(g p )

^.

Proof ofLemma4. Remarkthat,for

p

^large^enough,

∀ x ∈ (0, 1], x ^−k |f p (x)| |u p (x)| ≤ C k

|u(x)| + r(x)

where

r

îs â^bounded ^Borel ^function ôn

(0, 1]

^. ^The ûpper^bound îs ân întegrable^function ôn

(0, 1]

^,^so^that^the^dominatedconvergencetheoremyields

Z 1 0

x ^−k f p (x) u p (x) dx → Z 1

0 x ^−k f (x) u(x) dx

as

p → ∞

^,^which^proves^the^rst^part^of^the^lemma.

Since

v

^is ^bounded ^on

[1, +∞)

^,

(g p v p )

^converges ^uniformly ^to

gv

^on

[1, +∞)

^. ^The ^function

x 7→ x ^−k

^being^integrable^on

[1, +∞)

^,^the^dominatedconvergencetheoremyields

Z +∞

1 x ^−k g p (x) v p (x) dx → Z +∞

1 x ^−k g(x) v(x) dx

as

p → ∞

^,^which^concludes^the^proof^of^Lemma^4.

Proof ofLemma5. Remarkthat

1 p + 1 − 1

p = O 1

p ²

and

1 p + 2 − 2 p + 1 + 1

p = O 1

p ³

toobtaintheresult.

Proof ofLemma6. Itisclearthatforall

x ∈ (0, 1]

^,

f p (x) → e ^−1/x

^as

p → ∞

^.

Inorderto provethat

(f p ) ∈ P

^,^let^us^rewrite

f p (x)

^as

f p (x) = σ p ϕ p (x) ψ p (x)

^where

σ p =

1 − 1

p −α−1

, ϕ p (x) =

1 + 1

(p − 1)x −α−2

, ψ p (x) =

1 − 1

(p − 1)x + 1 p−1

,

forall

x ∈ (0, 1]

^,^and^prove^that

(σ p ) ∈ P 0

^,

(ϕ p ) ∈ P −1

^and

(ψ p ) ∈ P

^. ^First,^note^that

σ p = 1 + α + 1

p + (α + 1)(α + 2) 2

1 p ² + o

1 p ²

sothat thecollectionofconstantfunctions

(σ p )

^liesⁱⁿ

P 0

^. ^Second,^we^have

∀ p > 1, ∀x ∈ (0, 1], |ϕ p (x)| ≤ 1 ≤ x ⁻¹ .

⁽¹⁵⁾

(20)

Moreover,

[ϕ p+1 − ϕ p ](x) = ϕ p (x)

"

1 − 1 p

−α−2

1 − x px + 1

α+2

− 1

# ,

andsince

∀ x ∈ (0, 1]

^,

x/(px + 1) ≤ 1/p

^,^Tâylorêxpansions^yield,ûniformlyⁱⁿ

x ∈ (0, 1]

^,

[ϕ p+1 − ϕ p ](x) = ϕ p (x)

α + 2 p(px + 1) + O

1 p ²

.

Itfollowsthat thereexistsapositiveconstant

C ⁽¹⁾

^such ^that^for

p

^large^enough,

p ² |ϕ p+1 − ϕ p |(x) ≤ C ⁽¹⁾ x ⁻¹ .

⁽¹⁶⁾

Third,let

x ∈ (0, 1]

^,ând^considerâ^pointwise^Taylorêxpansionôf

ϕ p

^to^get

ϕ p (x) = 1 − α + 2

px + α + 2 p ² x

−1 + α + 3 2x

+ o

1 p ²

.

Using(15),(16)andapplyingLemma 5thereforeshowsthat

(ϕ p ) ∈ P −1

^.

Let

x ∈ (0, 1]

^,

k ≥ 0

^,

Ψ x (p) = (1 − 1/(px + 1)) ^p

^,^so^that

ψ p (x) = Ψ x (p−1)

^. ^Routinecalculations showthat

Ψ x (p)

^is^a^positivenon-increasingfunctionof

p

^. Consequently,forallsucientlylarge

p

^and^for^all

x ∈ (0, 1]

^,

ψ p (x) ≤ ψ k+1 (x)

^. ^Remarking^that

ψ k+1 (x) ≤ k ^k x ^k

^for^all

x ∈ (0, 1]

^,^it

followsthat

∀ k ≥ 0, ∃ p k ≥ 1, ∃ C k ≥ 0, ∀ p ≥ p k , ∀ x ∈ (0, 1], |ψ p (x)| ≤ C k x ^k .

Estimating an endpoint with high order moments Stéphane Girard, Armelle Guillou, Gilles Stupfler

HAL Id: inria-00596979

https://hal.inria.fr/inria-00596979

Submitted on 30 May 2011

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