Estimating an endpoint with high order moments in the Weibull domain of attraction

(1)

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Estimating an endpoint with high order moments in the

Weibull domain of attraction

Stephane Girard, Armelle Guillou, Gilles Stupfler

To cite this version:

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

in the Weibull domain of attraction. Statistics and Probability Letters, Elsevier, 2012, 82 (12),

pp.2136-2144. �10.1016/j.spl.2012.07.005�. �hal-00648435v2�

(2)

in the Weibull domain of attra tion

Stéphane Girard

(1)

, Armelle Guillou

(2)

&GillesStuper

(2)

(1)

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

Montbonnot,38334Saint-Ismier edex,Fran e

(2)

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

67084Strasbourg edex,Fran e

Abstra t. Wepresentamethod forestimatingtheendpointofaunidimensionalsamplewhen

thedistributionfun tionbelongstotheWeibull-maxdomainofattra tion. Theapproa hrelieson

transformingthevariableofinterestandthenusinghighordermomentsofthepositivevariable

ob-tainedthisway. Itisassumedthattheorderofthemomentsgoestoinnity. Wegive onditionson

therateofdivergen etogettheweakandstrong onsisten yaswellastheasymptoti normalityof

theestimator. Thegoodperforman eoftheestimatorisillustratedonsomenitesamplesituations.

AMS Subje t Classi ations: 62G32,62G05.

Keywords: Endpointestimation,highordermoments, onsisten y,asymptoti normality.

1 Introdu tion

Let

(X

1 , . . . , X

n

)

beindependent opiesofarandomvariable

X

,wherethedistribution of

X

has anite rightendpoint

θ

,with

θ

beingunknown. Weareinterestedin estimating

θ

. Re entwork onendpointestimation in ludes aBayesianlikelihood approa h(Hall andWang, 2005), ensored

likelihood estimators(Liet al., 2011a)and theempiri allikelihood method (Li et al., 2011b). In

Girardetal. (2011),anewestimatorof

θ

, basedupontheuseof high ordermomentsof the

X

k

, is introdu ed. From a pra ti al point of view, taking high order moments gives (exponentially)

moreweightto the

X

k

loseto

θ

;theoriginalideaisduetoGirardandJa ob(2008). Athorough study of the estimator is arried out when

X

is a positive random variable and its distribution fun tion belongs totheWeibull max-domainofattra tion. Inthispaper,weaddresstheproblem

of estimating the endpoint

θ

with high order moments when the positivity assumption on

X

is dropped. We annot use momentsof the variable of interest

X

, sin e

|X|

ould have aninnite

(3)

mean. Toover ome this problem, it an be noted that the random variable

e

X

has a bounded support

0, e

θ

. Moreover, letting

µ

p

_n

:= E e

p

n

X

,

p

n

→ ∞

be the

p

n

th order moment of

e

X

,

yields,forall

u

≥ 1

,

µ

p

n

/µ

p

n

+u

→ e

−uθ

as

n

→ ∞

(seeLemma1intheAppendix). Forall

a > 0

, itfollowsthat

Θ

n

:=

1 a

log

µ

p

n

µ

p

n

+1

− log

_µ

(a+1)p

n

µ

(a+1)p

n

+a+1

= θ(1 + o(1)).

(1)

Wethereforeintrodu ean estimatorusinghigh order momentsofthevariable

e

X

. Repla ingthe

truemoment

µ

p

n

withitsempiri al ounterpart

µ

b

p

n

in theexpressionof

Θ

n

yields

b

θ

n

:=

1 a

log

b

µ

p

n

b

µ

p

n

+1

− log

_µ

_b

(a+1)p

n

b

µ

(a+1)p

n

+a+1

where

(p

n

)

isapositive,nonrandomsequen esu hthat

p

n

→ ∞

,

a > 0

and

b

µ

p

n

:=

1 n

n

X

i=1

e

p

n

X

i

is the lassi al moment estimatorof

µ

p

n

. It is shown in Se tion 2 that

θ

b

n

is onsistent without any parametri assumption on the distribution of

X

. Moreover, we state and prove that

θ

b

n

is asymptoti allyGaussianwhenthedistributionfun tionof

X

belongstotheWeibullmax-domainof attra tion. SomesimulationsareproposedinSe tion3toillustratethee ien yofourestimator,

and to ompare it with estimators of the endpoint estimation literature. Auxiliary results are

postponedtotheAppendix.

2 Main results

Letusrststatethe onsisten yoftheestimator. Theonlyassumptionis

(A

0 ) X

hasanite rightendpoint

θ

.

Theorem1. If

(A

0 )

holds and

n µ

(a+1)p

n

/e

(a+1)p

n

θ

→ ∞

as

n

→ ∞

,then

b

θ

n

P

−→ θ

as

n

→ ∞

.

Proof. Werstshowthat,provided

n µ

p

n

/e

p

n

θ

→ ∞

,thehighordermoment

µ

p

n

anberepla ed

byitsempiri al ounterpart

µ

b

p

n

in (1). Forall

ε > 0

,Chebyshev'sinequalityleadsto

P

b

µ

p

n

µ

p

n

− 1

> ε

≤

_ε

1

2 Var (b

µ

p

n

)

µ

2 p

n

≤

_nε

1

2 µ

2p

n

µ

2 p

n

,

where

µ

2p

n

µ

p

n

= e

p

n

θ

µ

2p

n

/e

2p

n

θ

µ

p

n

/e

p

n

θ

≤ e

p

n

θ

(2) andtherefore

P

µ

b

p

n

µ

p

n

− 1

> ε

≤

_ε

1

2 e

p

n

θ

n µ

p

n

→ 0

as

n

→ ∞

. Asa onsequen e,

µ

b

p

_n

/µ

p

_n

P

−→ 1

as

n

→ ∞

. Sin e

µ

(a+1)p

n

/e

(a+1)p

n

θ

≤ µ

p

n

/e

p

n

θ

, it follows that

n µ

p

n

/e

p

n

θ

→ ∞

. Lemma 1 thus yields

n µ

p

n

+1

/e

(p

n

+1)θ

→ ∞

and

n µ

(a+1)p

n

+a+1

/e

((a+1)p

n

+a+1)θ

→ ∞

as

n

→ ∞

. Consequently, from(1),

θ

b

n

= Θ

n

+ o

P

(1)

. Lemma1thenentails

θ

b

n

P

(4)

Underasomewhatstronger onditionoftherateof divergen e of

(p

n

)

,astrong onsisten yresult anbeestablishedfor

θ

b

n

:

Theorem2. If

(A

0 )

holds and

1 log n

n µ

(a+1)p

n

e

(a+1)p

n

θ

→ ∞

as

n

→ ∞

,then

θ

b

n

a.s.

−→ θ

as

n

→ ∞

.

Proof. Theresultbeingobviouswhen

P

(X = θ) = 1

,letusassumethat

P

(X = θ) < 1

. Westart byshowingthat

1 log n

n µ

p

n

e

p

n

θ

→ ∞ ⇒

b

µ

p

n

µ

p

n

a.s.

−→ 1

as

n

→ ∞.

(3)

Tothisend,let

Y

i

= e

X

i

−θ

. Then

|Y

p

n

i

− E(Y

p

n

i

)

| ≤ 1

a.s. and

b

µ

p

n

− µ

p

n

e

p

n

θ

=

1 n

n

X

i=1

Y

p

n

i

− E(Y

p

n

i

)

isameanofbounded, entered,independentandidenti allydistributedrandomvariables. Dening

τ

n

:= ε

n µ

p

n

e

p

n

θ

and

λ

n

:= ε

µ

p

n

e

p

n

θ

1 Var(Z

p

n

1 )

= ε

µ

p

n

e

p

n

θ

e

2p

n

θ

µ

2p

n

− µ

2 p

n

,

Bernstein'sinequality(seeHoeding,1963)gives,forall

ε > 0

,

P

b

µ

p

n

µ

p

n

− 1

> ε

= P

µ

b

p

n

− µ

p

n

e

p

n

θ

> ε

µ

p

n

e

p

n

θ

≤ exp

−

_{2(1 + λ}

τ

n

λ

n

/3)

.

Notethatsin e

e

X−θ

∈ [0, 1]

a.s.,wehave

µ

p

n

/e

p

n

θ

→ P(X = θ)

as

n

→ ∞

. Then,from(2),

1 λ

n

≤

1 ε

h

1 ₋

µ

p

n

e

p

n

θ

i

→

1 − P(X = θ)

_ε

> 0

as

n

→ ∞

,andtherefore,forsu ientlylarge

n

,thereexistsa onstant

C

ε

> 0

su hthat

P

b

µ

_µ

p

_p

n

_n

− 1

> ε

≤ exp

−C

ε

n µ

p

n

e

p

n

θ

.

Borel-Cantelli's lemma thus yields

µ

b

p

n

/µ

p

n

a.s.

−→ 1

as

n

→ ∞

. Using on e again the inequality

µ

(a+1)p

n

/e

(a+1)p

n

θ

≤ µ

p

n

/e

p

n

θ

,Lemma1and(3), theresultisnowstraightforward.

Letusnowestablishtheasymptoti distributionofourestimator. Tothisend,additional

assump-tionsareintrodu edonthesurvivalfun tion

F = 1

− F

of

X

:

(A

1 )

∀ x < θ

,

F (x) = (θ

− x)

α

_L((θ

− x)

−1

₎

where

θ

∈ R

,

α > 0

and

L

is a slowly varying fun tionat innity,i.e. su hthat

L(ty)/L(y)

→ 1

as

y

→ ∞

forall

t > 0

.

(A

2 )

∀ x ≥ 0

,

L(x) = c exp

R

x

1 η(t) t

−1

_dt

, where

c > 0

and

η

is abounded fun tion tending to

0

at innity, ontinuouslydierentiableon

(0,

∞)

, ultimatelymonotoni and nonidenti ally

0

, su h that

|η

′

_|

isregularlyvaryingandthere exists

ν

≤ 0

with

x η

′

_(x)/η(x)

_{→ ν}

as

x

→ ∞

.

Itiswell-knownthat

(A

1 )

holdsifandonlyif

F

belongstotheWeibullmax-domainofattra tion, seeFisherand Tippett (1928)andGnedenko(1943).

(A

2 )

istheKaramatarepresentationforthe normalizedslowlyvaryingfun tion

L

, seeBinghamet al. (1987),p. 15. Under

(A

2 )

,thefun tion

(5)

(1987),paragraph 1.4.2,and the fun tion

x

7→ x |η

′

_(x)

_|

is regularlyvarying with index

ν

. Inthe extreme-valueframework,

ν

isreferredtoasthese ondorderparameterand

(A

2 )

isase ondorder ondition.

We annowstatetheasymptoti normalityof

θ

b

n

: Theorem3. Assume

(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

_d

−→ N (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. Let us remark that

v

n

b

θ

n

− θ

= v

n

b

θ

n

− Θ

n

+ v

n

(Θ

n

− θ)

and fo us on therandom

term. Ourgoalistoestablishthat

v

n

b

θ

n

− Θ

n

_d

−→ N (0, V (α, a))

as

n

→ ∞

. Tothisend,using thedelta-method,itisenoughtoprovethatthesequen eof randomvariables

ξ

n

:=

e

−aθ

a

p

V (α, a)

v

n

e

ab

θ

n

− e

aΘ

n

onvergesindistributiontoastandardGaussianrandomvariable. Noti ethatthe hangeofvariable

z = (θ

_{− x)}

−1

yields

µ

p

= p

Z

θ

−∞

e

px

_{F (x) dx = p}

−α

_e

pθ

_{L(p) [Γ(α + 1) + I}

1 ε

1 (p) + I

2 ε

2 (p)]

(4) where

I

1 =

Z

∞

1 z

α

e

−z

dz

,

I

2 =

Z

1

0 z

α

e

−z

dz

and

ε

1 (p)

=

1 I

1 Z

1

0 e

−1/z

_z

−α−3

L

1 (pz)

L

1 (p)

dz

_{− 1,}

L

1 (z) =

z L(z),

ε

2 (p)

=

1 I

2 Z

∞

1 e

−1/z

z

−α−1

L

2 (pz)

L

2 (p)

dz

_{− 1,}

L

2 (z) =

L(z)

z

.

Using(4)togetherwithLemma2i) entails

ξ

n

= e

−(a+1)θ

µ

p

n

+1

u

n, a

∆

(1)

n

+ ∆

(2)

n

(1 + o(1))

(5) with

u

n, a

=

1 a Γ(α + 1)

s

1 V (α, a)

p

α

n

v

n

e

p

n

θ

L(p

n

)

,

∆

(1)

n

=

b

µ

p

n

b

µ

p

n

+1

−

µ

p

n

µ

p

n

+1

b

µ

(a+1)p

n

+a+1

b

µ

(a+1)p

n

,

∆

(2)

n

=

b

µ

(a+1)p

n

+a+1

b

µ

(a+1)p

n

−

µ

(a+1)p

n

+a+1

µ

(a+1)p

n

µ

p

n

µ

p

n

+1

.

Rewriting

∆

(1)

n

and

∆

(2)

n

yields

ξ

n

= u

n, a

ζ

n

(1)

+

e

−(a+1)θ

b

µ

(a+1)p

n

+a+1

b

µ

(a+1)p

n

· µ

p

n

+1

b

µ

p

n

+1

− 1

ζ

n

(2)

+

_µ

(a+1)p

n

b

µ

(a+1)p

n

− 1

ζ

n

(3)

(1 + o(1)),

(6)

where,setting

ν

p

= b

µ

p

− µ

p

,

ζ

n

(1)

= ζ

n

(2)

+ ζ

n

(3)

,

with

ζ

(2)

n

= ν

p

n

−

µ

p

n

µ

p

n

+1

ν

p

n

+1

,

and

ζ

(3)

n

= e

−(a+1)θ

µ

p

n

µ

(a+1)p

n

+a+1

µ

2 (a+1)p

n

−ν

(a+1)p

n

+

µ

(a+1)p

n

µ

(a+1)p

n

+a+1

ν

(a+1)p

n

+a+1

.

Inviewoftheabove onsisten yresults,itfollowsthat

ξ

n

= u

n, a

h

ζ

n

(1)

+ o

P

ζ

n

(2)

+ o

P

ζ

n

(3)

i

(1 + o(1)),

anditisthereforesu ientto showthat

u

n, a

ζ

n

(1)

d

−→ N (0, 1),

(6a)

u

n, a

ζ

n

(2)

d

−→ N (0, C

2 ),

(6b)

u

n, a

ζ

n

(3)

d

−→ N (0, C

3 ),

(6 )

where

C

2

and

C

3

aresuitable onstants. Letusthenwrite

ζ

(1)

n

=

n

X

k=1

S

n, k

,where

S

n, k

=

1 n

h

e

p

n

X

k

_{, e}

(p

n

+1)X

k

_{, e}

(a+1)p

n

X

k

_{, e}

[(a+1)p

n

+a+1]X

k

i

A

n

,

A

n

=

[a

n, 0

, a

n, 1

, a

n, 2

, a

n, 3

]

t

,

a

n, 0

=

1,

a

n, 1

=

−

µ

p

n

µ

p

n

+1

,

a

n, 2

=

−e

−(a+1)θ

µ

p

n

µ

(a+1)p

n

+a+1

µ

2 (a+1)p

n

,

a

n, 3

=

e

−(a+1)θ

µ

p

n

µ

(a+1)p

n

.

Sin ethe

S

n, k

,

1 ≤ k ≤ n

,areindependent,identi ally distributedand enteredrandomvariables, weshallprovethat

E

_|S

_{n, 1}

_|

3 √

_{n [Var (S}

n, 1

)]

3/2

→ 0

as

n

→ ∞

,anduseLyapounov'stheorem(seee.g.Billingsley,1979,p.312)toobtaintheasymptoti normalityof

ζ

(1)

n

.

Anequivalentof

Var (S

n, 1

)

isobtainedbyusing(4)andapplyingLemma2toget

Var (S

n, 1

) = a

2 Γ

2 (α + 1) V (α, a)

1 n

e

2p

n

θ

p

−α−2

n

L(p

n

) (1 + o(1)).

To ontrol

E

|S

n, 1

|

3

,introdu ing

Y = X

− θ

, Hölder'sinequalityyields

E

_|S

_{n, 1}

_|

3 n

−3

_e

3p

n

θ

≤ 4 E

e

p

n

Y

a

n, 0

+ a

n, 1

e

θ

e

Y

3 +

4 E

_e

(a+1)p

n

Y

h

a

n, 2

e

ap

n

θ

+ a

n, 3

e

(ap

n

+a+1)θ

e

(a+1)Y

i

3

(7)

Let us remark that

Y

has survival fun tion

G

dened by

G(y) = (

−y)

α

_{L (}

_−y)

−1

, for all

y

∈

(

−∞, 0)

. Setting

H

n, 0

(z) = 1,

H

n, 1

(z) =

−αz,

H

n, 2

(z) =

−e

ap

n

θ

µ

p

n

µ

(a+1)p

n

1 _{− z}

a+1

1 _{− z}

,

H

n, 3

(z) = α e

ap

n

θ

µ

p

n

µ

(a+1)p

n

,

somemoreeasy omputations showthat there exist twosequen es of Borelfun tions

(χ

n, 1

)

and

(χ

n, 2

)

uniformly onvergingto0on

[0, 1]

su hthat forall

z

∈ [0, 1]

,

a

n, 0

+ a

n, 1

e

θ

z

=

H

n, 0

(z)(1

− z) +

H

n, 1

(z) + χ

n, 1

(z)

p

n

,

a

n, 2

e

ap

n

θ

+ a

n, 3

e

(ap

n

+a+1)θ

z

a+1

=

H

n, 2

(z)(1

− z) +

H

n, 3

(z) + χ

n, 2

(z)

p

n

.

ApplyingLemma4twi eentails

E

|S

n, 1

|

3 = O n

−3

_e

3p

n

θ

_p

−α−3

n

L(p

n

)

. Lyapounov'stheoremthen

gives(6a). Proofsof(6b)and(6 )arethensimilar.

Letusnowfo usonthenonrandomterm

v

n

(Θ

n

− θ)

. Re alling(4) andletting

τ (p, u) :=

I

1 [ε

1 (p)

− ε

1 (p + u)] + I

2 [ε

2 (p)

− ε

2 (p + u)]

Γ(α + 1) + I

1 ε

1 (p + u) + I

2 ε

2 (p + u)

,

onehas

∀ u ≥ 1,

_µ

µ

p

p+u

= e

−uθ

1 +

u

p

α

exp

−

Z

p+u

p

η(t)

t

dt

[1 + τ (p, u)].

Letusnotethat

Z

p+u

p

η(t)

t

dt = O

|η(p)|

p

andapplyLemma 2toget

∀ u ≥ 1,

_µ

µ

p

p+u

= e

−uθ

1 +

u

p

α

+ O

|η(p)|

p

.

Itisthen learthat

Θ

n

= θ + O

|η(p

n

)

|

p

n

.

Theresultfollowsfrom Slutsky'slemma.

Letusnotethatthe hoi eofanoptimal valuefor

a

byminimizationof

V (α, a)

isadi ulttask sin eitdepends ontheunknownvalueof

α

. One anobserveonFigure 1that,for

α

≤ 2

,

V (α,

·)

isade reasingfun tionandthuslargevaluesof

a

should bepreferred.

Asfarastherateof onvergen e

v

n

oftheestimatoris on erned,notethatuptoaslowlyvarying fa tor,onehas

v

n

=

√

n p

−α/2+1

n

, where

(p

n

)

satises

n p

−α

n

→ ∞

and

n p

−α

n

η

2 (p

n

)

→ 0

. Weshall onsiderthe ases

α

≥ 2

and

α < 2

separately:

(8)

1. If

α

≥ 2

, then the smaller

p

n

is, the higher

v

n

is. The onstraint on

(p

n

)

is therefore the ondition

n p

−α

n

η

2 (p

n

)

→ 0

. Sin e

|η|

is regularly varying with index

ν

, this ondition is essentially

n p

2ν−α

n

→ 0

: thesmallestpossiblesequen e

(p

n

)

satisfyingthis requirementhas order

n

1/(α−2ν)

. Consequently,

(v

n

)

hasorder

n

(1−ν)/(α−2ν)

.

2. If now

α < 2

, then the rate

(v

n

)

in reasesas

(p

n

)

in reases: the onstraint on

(p

n

)

is the ondition

n p

−α

n

→ ∞

. Thelargestpossiblesequen e

(p

n

)

satisfyingthis onditionhasorder

n

1/α

,whi h yieldsarate

(v

n

)

withorder

n

1/α

.

Hen e,theestimatorof Aarssen andde Haan(1994)and ourestimatoressentiallyhavethe same

rateof onvergen e. Moreover,sin etherateof onvergen eofthemaximumestimatoris

n

1/α

(see

deHaanandFerreira,2006), weseethat in the ase

α < 2

, therateisthesameastheoneofthe maximum,andinthe ase

α

≥ 2

,therateisfasterthantheoneofthemaximum. Thethreeabove mentionedestimatorsare omparedonnitesamplesituationsin thenextse tion.

3 Numeri al illustration

Here,weexaminetheperforman esofourestimatorby onsideringtwodierentmodels. Therst

onehassurvivalfun tion

∀ x < 0, F (x) =

1 + (

−x)

−τ

1 −τ

2

(7)

with

τ

1 , τ

2 > 0

, that is,

X =

−1/Z

where

Z

has a Burr(

1, τ

1 , τ

2

) type XII distribution as in Beirlantet al. (2004). Here

(A

1 )

and

(A

2 )

hold,with

θ = 0

,

α = τ

1 τ

2

and

ν =

−τ

1

.

These ondonehassurvivalfun tion

∀ x < 0, F (x) =

Z

∞

log(1−1/x)

λ

2 t e

−λt

dt

(8)

with

λ > 0

,whi h istantamountto

X =

−1/(e

Z

_{− 1)}

where

Z

isGamma(

2, λ

)distributed. Some umbersome omputations showthat

(A

1 )

and

(A

2 )

holdwith

θ = 0

,

α = λ

and

ν = 0

.

Ea hofthesemodelsis onsideredwithdierentsetsofparameters,seetherst olumnofTable1.

The power

p := p

n

is hosen to vary a ross the set

P = {5, 10, 15, . . . , 300}

, and a set

A =

{0.1, 0.4, 0.7, . . . , 25}

of dierentvaluesof

a

istested. Inea hsituation,

N = 1000

repli ationsof asamplewithsize

n = 500

aregeneratedandtheaverage

L

1 ₋

error

E(p, a) =

1 N

N

X

j=1

|ε(j, p, a)| ,

where

ε(j, p, a) = b

θ

(j, p, a)

− θ

is omputed, with

θ

b

(j, p, a)

being the estimator omputed on the

j

th repli ation with

(p, a)

∈

P × A

and an endpoint

θ = 0

. Then, the optimal values of

p

and

a

are retained:

(p

⋆

_{, a}

⋆

_{) =}

argmin

{E(p, a), (p, a) ∈ P × A}

. Thesamepro edure isappliedtotheextreme-valuemoment es-timatorofAarssenanddeHaan(1994),whi hdependsonaparameter

k

∈ {2, 3, . . . , n − 1}

. The

(9)

E(p

⋆

_{, a}

⋆

₎

is displayed. Let us noti e that, in all the onsidered situations, our estimator yields

slightlybetter(optimal)resultsthanthemaximumandtheextreme-valuemomentestimator.

Tofurther omparethebehavioroftheestimatorsintheoptimal ase,boxplotsoftheasso iated

errors

ε(j, p

⋆

_{, a}

⋆

₎

aredisplayedonFigure23. Unlikethemaximumestimator,ourestimatordoes

notalwaysunderestimatetheendpoint. Moreover,theerrorasso iatedtoourestimatorissmaller

thantheerrorofthemaximum. Besides,thevarian eofourestimatorissimilarto theoneofthe

maximum,anditissmallerthantheoneoftheextreme-valuemomentestimator.

OnFigure4,we omparethefun tions

E

asso iatedtothethreeestimators. Onmodel (7),while the error asso iated to the extreme-value moment estimator appears to be very sensitive to the

hoi e of

k

,theerrorasso iatedto ourestimatorisstable foralargepanel ofvaluesof

p

n

and

a

. Resultsaresimilarintheother onsidered ases.

A knowledgements

Theauthorsare indebted to theanonymousreferee forhis/herhelpful ommentsand suggestions

thathave ontributedtoanimprovedpresentationoftheresultsofthis paper.

Referen es

Aarssen,K., deHaan,L., 1994. Onthe maximallife span ofhumans. Math. Popul. Stud. 4(4),

259281.

Beirlant,J., Goegebeur, Y., Segers,J., Teugels, J.,2004. Statisti sof Extremes,JohnWiley and

Sons.

Billingsley,P., 1979. Probabilityand measure,JohnWileyandSons.

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

UniversityPress.

Fisher,R.A.,Tippett, L.H.C.,1928. Onestimating ofthefrequen ydistributions ofthelargestor

smallestmemberofasample. Pro . Camb. Phil. So . 24,180190.

Girard,S.,Guillou,A.,Stuper,G.,2011. Estimatinganendpointwithhighordermoments. Test,

toappear. DOI:10.1007/s11749-011-0277-8.

Girard, S.,Ja ob, P., 2008. Frontierestimation via kernel regressionon high power-transformed

data. J.MultivariateAnal. 99(3),403420.

Gnedenko,B.V.,1943. Surladistributionlimitedumaximumd'unesériealéatoire. Ann. ofMath.

(10)

Hall,P.,1982. Onestimatingtheendpointofadistribution. Ann. Statist. 10(2),556568.

Hall,P.,Wang,J.Z., 2005. Bayesianlikelihood methodsforestimatingtheendpointofa

distribu-tion. J.Roy. Statist. So . Ser. B67(5),717729.

Hoeding, W., 1963. Probability inequalities for sums of bounded random variables. J. Amer.

Statist. Asso . 58,1330.

Li,D.,Peng,L.,Xu,X.,2011a. Bias redu tionforendpointestimation. Extremes14,393412.

Li, D., Peng, L, Qi, Y., 2011b. Empiri al likelihood onden e intervals for the endpoint of a

distributionfun tion. Test 20,353366.

4 Appendix: Auxiliary results

The rst two results are analoguesof Lemmas 1 and 2 in Girard et al. (2011), as well astheir

proofs,whi h areomitted.

Lemma1. If

(A

0 )

holds, thenfor all

u

≥ 1

,one has

µ

p

/µ

p+u

→ e

−uθ

as

p

→ ∞

.

Lemma2. Assumethat

(A

1 )

holds anddene

ε

1

and

ε

2

asinthe proof ofTheorem 3. Then,for all

i = 1, 2

,

(i)

ε

i

(p)

→ 0

as

p

→ ∞

.

Moreover, if

L

satises

(A

2 )

,thenfor all

i = 1, 2

and

u, v

≥ 1

, (ii)

ε

i

(p + u)

− ε

i

(p) = O(

|η(p)|/p)

,

(iii)

p

2 _(ε

i

(p + u + v)

− ε

i

(p + v)

− [ε

i

(p + u)

− ε

i

(p)])

→ 0

as

p

→ ∞

.

Thenextlemmais ate hni alresultwhi hshallbeusefulin theproofof Lemma4below. Itis a

simple onsequen eofLemma2:

Lemma3. Assumethat

(A

1 )

holds. Then, as

p

→ ∞

,

∀ d ≥ 0,

p

Z

θ

−∞

e

px

(θ

_{− x)}

d

F (x) dx = p

−α−d

e

pθ

L(p) Γ(α + d + 1) (1 + o(1)).

Proofof Lemma3. Rewritetheleft-hand sideasin (4)andapplyLemma2i).

Thenallemmaofthisse tionprovidesanasymptoti boundofthethird-ordermomentsappearing

intheproofofTheorem3.

Lemma4. Let

m

∈ N

,

(H

n, j

)

,

0 ≤ j ≤ m

be sequen es of Borel uniformlybounded fun tionson

(0, 1)

and

(p

n

)

bearealsequen etendingtoinnity. Introdu e

∀ z ∈ (0, 1),

h

n

(z) =

m

X

j=0

H

n, j

(z)

p

j

n

(1

_{− z)}

m−j

,

(11)

andlet

Y

bea random variable withsurvival fun tion

G

denedby

∀ y ∈ (−∞, 0),

G(y) = (

_−y)

α

L (

_−y)

−1

where

α > 0

and

L

isaslowly varyingfun tion atinnity. Then

E

e

p

n

Y

_h

n

e

Y

3 = O p

−α−3m

n

L(p

n

)

.

Proofof Lemma4. Hölder'sinequalityyields

E

e

p

n

Y

h

n

e

Y

3 ≤ (m + 1)

2 





m

X

j=0

1 p

3j

n

sup

[0, 1]

n∈N\{0}

|H

n, j

|

3 E

e

p

n

Y

(1

_{− e}

Y

)

m−j

3 



 .

Itisenoughtoshowthat

∀ j ∈ {0, . . . , m}, E

e

p

n

Y

₍₁

− e

Y

₎

m−j

3 _{= O}

_p

−α−(3m−3j)

n

L(p

n

)

. An

integrationbypartsgives

E

_e

p

n

Y

₍₁

− e

Y

)

m−j

3 =

Z

0 −∞

d

dy

e

3p

n

y

₍₁

− e

y

)

3m−3j

G(y) dy.

If

(s

n

)

isarealsequen etendingtoinnity,

ϕ

isapositiveboundedfun tionon

(

−∞, 0)

and

β

≥ 0

, bywriting

∀ δ > 0,

Z

0 −∞

e

s

n

y

₍

−y)

β

_{ϕ(y) dy =}

Z

0 −δ

e

s

n

y

₍

−y)

β

_{ϕ(y) dy}







1 +

Z

−δ

−∞

e

s

n

y

(

−y)

β

_{ϕ(y) dy}

Z

0 −δ

e

s

n

y

₍

−y)

β

ϕ(y) dy







,

itisreadilyshownthat

∀ δ > 0,

Z

0 −∞

e

s

n

y

(

_−y)

β

ϕ(y) dy =

Z

0 −δ

e

s

n

y

(

_−y)

β

ϕ(y) dy (1 + o(1)).

(9)

Sin e

y/(1

− e

y

₎

_{→ −1}

as

y

→ 0

,weget,forall

ε > 0

, hoosing

δ = δ

0

small enough,

1 ₋

ε

2 ≤

Z

0 −δ

0 e

s

n

y

(1

− e

y

₎

d

_{G(y) dy}

Z

0 −δ

0 e

s

n

y

₍

−y)

d

_{G(y) dy}

≤ 1 +

ε

₂

.

Asa onsequen e,(9)yields,forallsu ientlylarge

n

,

1 _{− ε ≤}

Z

0 −δ

0 e

s

n

y

₍₁

− e

y

)

d

G(y) dy

Z

0 −∞

e

s

n

y

(

_−y)

d

G(y) dy

≤ 1 + ε.

(10)

Itonlyremainstouse(9)on eagainandtoapplyLemma3toobtain

Z

0 −∞

e

s

n

y

(1

− e

y

₎

d

_{G(y) dy = s}

−α−d−1

n

L(s

n

) Γ(α + d + 1) (1 + o(1)).

Repla ing in (10), it follows that

E

e

p

n

Y

₍₁

− e

Y

₎

m−j

3 _{= O}

_p

−α−(3m−3j)

n

L(p

n

)

, whi h estab-lishesLemma4.

(12)

Distribution Maximum

Aarssen&deHaan momentsestimator

−1/

Burr

(1, τ

1 , τ

2 )

(τ

1 , τ

2 ) = (1, 1)

2.0 · 10

−3

_2.1

_{· 10}

−3

_1.6

_{· 10}

−3

⇒ (α, ν) = (1, −1)

(τ

1 , τ

2 ) = (5/6, 6/5)

2.0 _{· 10}

−3

_2.0

· 10

−3

_1.6

· 10

−3

⇒ (α, ν) = (1, −5/6)

(τ

1 , τ

2 ) = (2/3, 3/2)

2.2 · 10

−3

_2.0

_{· 10}

−3

_1.7

_{· 10}

−3

⇒ (α, ν) = (1, −2/3)

(τ

1 , τ

2 ) = (1/2, 2)

2.3 _{· 10}

−3

_2.3

· 10

−3

_1.9

· 10

−3

⇒ (α, ν) = (1, −1/2)

−1/(exp(

Gamma

(2, λ))

− 1)

λ = 1

2.3 _{· 10}

−4

_2.0

· 10

−4

_1.9

· 10

−4

⇒ (α, ν) = (1, 0)

λ = 5/4

1.1 _{· 10}

−3

_9.2

· 10

−4

_8.5

· 10

−4

⇒ (α, ν) = (5/4, 0)

λ = 5/3

5.7 · 10

−3

_4.6

_{· 10}

−3

_4.1

_{· 10}

−3

⇒ (α, ν) = (5/3, 0)

λ = 5/2

3.1 _{· 10}

−2

_2.4

· 10

−2

_2.2

· 10

−2

⇒ (α, ν) = (5/2, 0)

Table1: Mean

L

1 ₋

errorsasso iatedtotheestimatorsin theeightsituations.

0

5

10

15

0.0

0.1

0.2

0.3

0.4

0.5

Figure 1: Graphsofthefun tions

a

7→ V (α, a)

. Solid line:

α = 1

, dashedline:

α = 5/4

, dashed-dottedline:

α = 5/3

,dotted line:

α = 2

,triangles:

α = 5/2

.

(13)

0.5

1.0

1.5

2.0

2.5

3.0

3.5 −0.010

−0.005

0.000

0.005

0.010

0.5

1.0

1.5

2.0

2.5

3.0

3.5 −0.015

−0.010

−0.005

0.000

0.005

0.010

0.015

0.020

0.5

1.0

1.5

2.0

2.5

3.0

3.5 −0.015

−0.010

−0.005

0.000

0.005

0.010

0.015

0.020

0.025

0.5

1.0

1.5

2.0

2.5

3.0

3.5 −0.015

−0.010

−0.005

0.000

0.005

0.010

0.015

Figure2: Boxplotsof

ε(j, p

⋆

_{, a}

⋆

₎

onmodel(7). Left: maximumestimator,middle: extreme-value

moment estimator, right: high order moments estimator. Top left:

(τ

1 , τ

2 ) = (1, 1)

; top right:

(τ

1 , τ

2 ) = (5/6, 6/5)

;bottomleft:

(τ

1 , τ

2 ) = (2/3, 3/2)

;bottomright:

(τ

1 , τ

2 ) = (1/2, 2)

.

0.5

1.0

1.5

2.0

2.5

3.0

3.5 −2e−3

−1.5e−3

−1e−3

−5e−4

0 5e−4

0.5

1.0

1.5

2.0

2.5

3.0

3.5 −0.008

−0.006

−0.004

−0.002

0.000

0.002

0.5

1.0

1.5

2.0

2.5

3.0

3.5 −0.020

−0.015

−0.010

−0.005

0.000

0.005

0.010

0.015

0.5

1.0

1.5

2.0

2.5

3.0

3.5 −0.05

0.00

0.05

0.10

Figure3: Boxplotsof

ε(j, p

⋆

_{, a}

⋆

₎

onmodel(8). Left: maximumestimator,middle: extreme-value

moment estimator, right: high order moments estimator. Top left:

λ = 1

; top right:

λ = 5/4

; bottom left:

λ = 5/3

;bottomright:

λ = 5/2

.

(14)

50 100 150 200 250 300 350 400 450

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

0.0035

0.0040

0.00215

0.00431

50

100

150

200

250

300

5

10

15

20

25 0.0017

0.0066

0.011

0.016

0.021 50 100 150 200 250 300 350 400 450

0.000

0.002

0.004

0.006

0.008

0.010 0.00574

0.0115

50

100

150

200

250

300

5

10

15

20

25 0.0041

0.015

0.026

0.037

0.048

Figure4: Comparisonofthethreeestimatorsonmodels(7)(top)and(8)(bottom). Left:

horizon-tally: threshold

k

, verti ally: error

E

, dashedline: maximumestimator,solid line: extreme-value momentestimator. Right: horizontally: parameter

p

n

,verti ally: parameter

a

, theerror

E

is rep-resentedwith shadesofgray, alongwith twolevel urves, respe tively orrespondingto themean