Asymptotic behaviour of extreme geometric quantiles and their estimation under moment conditions

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Asymptotic behaviour of extreme geometric quantiles and their estimation under moment conditions

Stéphane Girard, Gilles Stupfler

To cite this version:

Stéphane Girard, Gilles Stupfler. Asymptotic behaviour of extreme geometric quantiles and their

estimation under moment conditions. 2014. �hal-01060985�

and their estimation under moment onditions

Stéphane Girard

(1)

& GillesStuper

(2)

(1)

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

Montbonnot,38334Saint-Ismieredex,Frane

(2)

AixMarseilleUniversité,CNRS,EHESS,CentraleMarseille,GREQAMUMR7316,

13002 Marseille,Frane

Abstrat. A popular way to study the tail of a distribution is to onsider its extreme quantiles.

While this is a standard proedure for univariate distributions, it is harder for multivariate ones, pri-

marilybeausethere is nouniversally aepteddenition of what amultivariatequantileshould be. In

thispaper,wefousonextremegeometriquantiles. Theirasymptotisareestablished,bothindiretion

andmagnitude, under suitablemoment onditions,when thenorm of theassoiated index vetortends

to one. Inpartiular, it appears that if arandom vetorhas anite ovarianematrix, then the mag-

nitude of its extreme geometri quantilesgrows at a xed rate. We take advantageof these results to

deneanestimatorofextremegeometriquantilesofsuharandomvetor. Theonsistenyandasymp-

totinormalityoftheestimatorareestablishedandourresultsareillustratedonsomenumerialexamples.

AMS Subjet Classiations: 62H05,62G20,62G32.

Keywords: Extremequantile,geometriquantile,onsisteny,asymptotinormality.

1 Introdution

Let

X

^{bearandomvetorin}

R ^d

^{. Uptonow,severaldenitionsof}multivariatequantilesof

X

^havebeen

proposedinthestatistialliterature. WerefertoSering(2002)forareviewofvariouspossibilitiesforthis

notion. Here,wefousonthenotionofspatial orgeometriquantiles,introduedbyChaudhuri(1996),

whihgeneralisestheharaterisationof aunivariatequantileshowninKoenkerandBassett(1978). For

agivenvetor

u

^{belongingto theunitopenball}

B ^d

^of

R ^d

^{, where}

d ≥ 2

^{,ageometriquantilewithindex}

vetor

u

^{isanysolutionofthe}optimisationproblemdenedby

arg min

q∈R

^d

E(φ(u, X − q) − φ(u, X)),

⁽¹⁾

withthelossfuntion

φ : R ^d × R ^d → R, (u, t) 7→ k t k + h u, t i ,

^where

h· , ·i

^{istheusualsalarproduton}

R ^d

and

k · k

^{istheassoiatedEulideannorm. Notethat}

q(u) ∈ R ^d

^{possessesbothadiretionandmagnitude.}

Itanbeseenthatgeometriquantilesareinfatspeialasesof

M

^{quantilesintroduedbyBreklingand}

Chambers(1988)whihwerefurtheranalysedbyKolthinskii(1997). Besides,suhquantileshavevarious

strongproperties. First,thequantilewithindex vetor

u ∈ B ^d

^{isuniquewheneverthe}distribution of

X

isnotonentratedonasinglestraightlinein

R ^d

^{(seeChaudhuri,1996,orTheorem2.17inKemperman,}

1987). Seond,although theyarenotfullyaneequivariant,theyareequivariantunder any orthogonal

transformation (Chaudhuri, 1996). Third, geometri quantiles haraterise the assoiated distribution.

Namely, if two random variables

X

^and

Y

^{yield the same quantile funtion}

q

^{, then}

X

^and

Y

^have

thesame distribution(Kolthinskii, 1997). Finally, for

u = 0

^{, the well-known}

L ² −

^{geometri median is}

obtained,whihis thesimplestexampleofaentral quantile(seeSmall,1990). Wepointoutthat one

mayomputeanestimationofthegeometrimedian inaneientway,seeCardotetal. (2013).

Thesepropertiesmakegeometriquantilesreasonableandidateswhentryingtodenemultivariatequan-

tiles, whih is why their estimation was studied in several papers. We refer for instane to Chaud-

huri (1996),who establishedaBahadurexpansion forthe estimatorof geometri quantilesobtained by

solvingthe sample ounterpart of problem (1). Chakraborty(2001) then introdued atransformation-

retransformationproedure toobtain aneequivariantestimates of multivariatequantiles. This notion

wasextendedtoamultiresponselinearmodelbyChakraborty(2003). Reently,Dharetal. (2014)dened

amultivariatequantile-quantileplotusinggeometri quantiles. Conditionalgeometriquantilesanalso

be dened by substituting a onditional expetation to the expetation in (1). We referto Cadre and

Gannoun (2000) for the estimation of the onditional geometri median and to Cheng and de Gooijer

(2007)fortheestimationof anarbitraryonditional geometriquantile. Theestimationofaonditional

medianwhenthereisaninnite-dimensionalovariateisonsideredinChaouhandLaïb(2013).

Ourfousinthispaperisratheronextremegeometriquantiles,obtainedwhen

k u k → 1

^{. Thetheoryof}

univariateextremequantilesiswellestablished,seeforinstanethemonographbydeHaanandFerreira

(2006). Ontheontrary,thefewworksonextrememultivariatequantilesrelyonthestudyofextremelevel

setsoftheprobabilitydensityfuntionof

X

^{whenitisabsolutelyontinuouswithrespettotheLebesgue}

measure. We referforinstane to Cai et al. (2011)foran appliation tothe estimationof extremerisk

regions for nanial data orto Einmahl et al. (2013)who fous on the ase of bivariate distributions

withan appliation to insuranedata. Oneanalso analyse extremequantilesof multivariatedatasets

by seleting a univariate variable and onsidering the other variables as ovariates. This amounts to

estimating onditional univariate extreme quantiles: for a nite-dimensional ovariate, this problem is

onsidered in Daouia et al. (2013), the ase of a funtional ovariate being addressed in Gardes and

Girard(2012).

Inthisstudy,weprovideanequivalentofthediretionandmagnitudeoftheextremegeometriquantile

q(u)

^,

k u k → 1

^{undersuitablemomentonditions. Apartiularorollaryofourresultsisthatthemagnitude}

of the extreme geometri quantilesof a random vetor

X

^{having a nite ovarianematrix growsat a}

xed rate. Moreover,in this ase, themagnitude of the extreme geometri quantiles is asymptotially

haraterisedbytheovarianematrixof

X

^{. Thispropertyopensthedoortothedenitionofanextreme}

quantileestimator,whoseasymptotipropertiesarestudiedin thiswork.

Theoutlineofthepaperisasfollows. AsymptotipropertiesofgeometriquantilesarestatedinSetion2.

An appliation to the estimation of extreme geometri quantilesis given in Setion 3. Some examples

andillustrationsofourresultsarepresentedinSetion4. Setion5oersaoupleofonludingremarks.

ProofsaredeferredtoSetion 6.

2 Asymptoti behaviour of extreme geometri quantiles

Fromnowon,weassumethatthedistributionof

X

^{isnotonentratedonasinglestraightlinein}

R ^d

^and

non-atomi. Weshallreformulatetheoptimisationproblem(1)as

arg min

q∈ R

^d

ψ(u, q)

where

ψ : R ^d × R ^d → R, (u, q) 7→ E(φ(u, X − q) − φ(u, X))

^{anberewrittenas}

ψ(u, q) = E( k X − q k − k X k ) − h u, q i .

⁽²⁾

Chaudhuri(1996)provedthatinthisontext,thesolution

q(u)

^{of(1),namelythegeometriquantilewith}

indexvetor

u

^{,existsand isuniqueforevery}

u ∈ B ^d

^{. Dene furtherthat}

t/ k t k = 0

^if

t = 0

^;if

u ∈ R ^d

^is

suh thatthere isasolution

q(u) ∈ R ^d

^{to problem(1), thenthegradientof}

q 7→ ψ(u, q)

^mustbezeroat

q(u)

^,thatis

u + E

X − q(u) k X − q(u) k

= 0.

⁽³⁾

This ondition immediately entails that if

u ∈ R ^d

^{is suh that problem (1) has a solution}

q(u)

^{, then}

k u k ≤ 1

^{. Infat,weanproveastrongerresult:}

Proposition1. The optimisationproblem (1)has asolutionif andonly if

u ∈ B ^d

^.

Moreover,remarking that the funtion

ψ(u, · )

^{isstritly onvex,Chaudhuri (1996)proved thefollowing}

haraterisationofageometri quantile: forevery

u ∈ B ^d

^,

q(u)

^{isthesolutionofproblem(1)ifandonly}

ifitsatisesequation(3). Inpartiular,thisentailsthatthefuntion

G : R ^d → B ^d

^denedby

∀ q ∈ R ^d , G(q) = − E

X − q k X − q k

isaontinuousbijetion. Proposition2.6(iii)inKolthinskii(1997)showsthattheinverseofthefuntion

G

^{,i.ethegeometriquantilefuntion}

u 7→ q(u)

^{,is alsoontinuouson}

B ^d

^.

Inmostaseshowever,omputingexpliitlythefuntion

G

^{isahopelesstask,whihmakesitimpossible}

to obtain a losed-form expression for the geometri quantile funtion. It is thus of interest to prove

generalresultsaboutthegeometri quantile

q(u)

^{,espeiallyregardingitsdiretionand magnitude. Our}

rstmainresultfousesonthespeialaseofspheriallysymmetridistributions.

Proposition2. If

X

^{has aspheriallysymmetri} distributionthen:

(i) Themap

u 7→ q(u)

^{ommutes witheverylinearisometryof}

R ^d

^{. Espeially, the normofageometri}

quantile

q(u)

^{only dependson the normof}

u

^.

(ii) Forall

u ∈ B ^d

^{,the geometri quantile}

q(u)

^{has diretion}

u

^if

u 6 = 0

^and

q(0) = 0

^otherwise.

(iii) The funtion

k u k 7→ k q(u) k

^{isaontinuousinreasingfuntion on}

[0, 1)

^.

(iv) Itholdsthat

k q(u) k → ∞

^as

k u k → 1

^.

Although therst and third statement of Proposition 2annot be expeted to hold true for a random

variablewhihis notspheriallysymmetri,onemaywonderiftheseond andfourthstatement,namely

that a geometri quantile shares the diretion of its index vetorand that the norm of the geometri

quantile funtion tends to innity on the unit sphere, an be extended to the general ase. The next

result,whih examinesthebehaviourof thegeometri quantilefuntion nearthe boundaryof theopen

ball

B ^d

^{,providesananswertothis question.}

Theorem1. Let

S ^d−1

^{bethe unitsphereof}

R ^d

^.

(i) Itholdsthat

k q(v) k → ∞

^as

k v k → 1

^.

(ii) Moreover,if

v → u

^with

u ∈ S ^d−1

^and

v ∈ B ^d

^then

q(v)/ k q(v) k → u.

Theorem 1 shows twoproperties of geometri quantiles: rst, the norm of the geometri quantile

q(v)

with index vetor

v

^{diverges to innity as}

k v k ↑ 1

^{. In other words,} Proposition 2(iv) still holds for any distribution. This is arather intriguing property of geometri quantiles, sine it holds even ifthe

distributionof

X

^{hasaompatsupport. Arelatedpointis thefat that samplegeometriquantilesdo}

notneessarilyliewithin theonvexhullofthesample,seeBreklingetal. (2001)foraounter-example.

Seond,if

v → u ∈ S ^d−1

^{thenthe geometriquantile}

q(v)

^{has asymptotidiretion}

u

^. Proposition 2(ii) thusremainstrueasymptotiallyforanydistribution.

ItispossibletospeifytheonvergenesobtainedinTheorem1undermomentassumptions. Theorem2

providesarst-order expansionofthe diretion andof themagnitude ofan extremegeometri quantile

q(αu)

^{inthediretion}

u

^,where

u

^{isaunit vetorand}

α

^tendsto1.

Theorem2. Let

u ∈ S ^d−1

^.

(i) If

E k X k < ∞

^then

q(αu) − {k q(αu) k u + E(X − h X, u i u) } → 0

^as

α ↑ 1.

(ii) If

E k X k ² < ∞

^and

Σ

^{denotes the ovarianematrixof}

X

^then

k q(αu) k ² (1 − α) → 1

2 (tr Σ − u ^′ Σu) > 0

^as

α ↑ 1.

AsaonsequeneofTheorem2,itappearsthatif

X

^{hasaniteovarianematrix}

Σ

^{thenthemagnitude}

ofanextremegeometriquantileinthediretion

u

^{isdetermined(intheasymptotisense)by}

Σ

^{. Inother}

words,sinetheasymptotidiretion ofanextremegeometriquantileinthediretion

u

^isexatly

u

^by

Theorem1,it followsthat theextremegeometri quantilesof twoprobabilitydistributions whihadmit

thesameniteovarianematrixareasymptotiallyequivalent. Furthermore,weobservethat

k q(βu) k k q(αu) k =

1 − α 1 − β

1/2

(1 + o(1))

when

α → 1

^and

β → 1

^{. Inotherwords,givenanarbitraryextremegeometriquantile,oneandedue}

theasymptotibehaviourofeveryotherextreme geometriquantilesharingits diretion,independently

ofthedistribution. Thisisfundamentallydierentfromtheunivariateasewhendeduingthevalueofan

extremequantilefromanotheronerequirestheknowledgeoftheextreme-valueindexofthedistribution,

seedeHaanandFerreira(2006),Chapter 4. Ourresultsanatuallybeused todene aonsistentand

asymptotiallyGaussianestimatorofextremegeometriquantiles,asshowninSetion 3below.

3 An estimator of extreme geometri quantiles

Let

X 1 , . . . , X n

^beindependentrandomopiesofarandomvetor

X

^{havinganiteovarianematrix}

Σ

^.

ItfollowsfromTheorem2thatanyextremegeometriquantile

q(αu)

^of

X

^,with

α ↑ 1

^and

u ∈ S ^d−1

^an

beapproximatedby:

q eq (αu) := (1 − α) ^−1/2 1

2 (tr Σ − u ^′ Σu) 1/2

u.

⁽⁴⁾

Thisanbeusedtodeneanestimatoroftheextremegeometriquantilesof

X

^{: let}

X n = n ⁻¹ P n k=1 X k

bethesamplemeanand

Σ b n = 1 n

X n k=1

(X k − X n )(X k − X n ) ^′

bethe empirialestimatoroftheovarianematrix

Σ

^of

X

^{. Let further}

(α n )

^{beaninreasingsequene}

ofpositiverealnumberstendingto 1. Ourestimator

q b n (α n u)

^of

q(α n u)

^isthen

b

q n (α n u) = (1 − α n ) ^−1/2 1

2

tr Σ b n − u ^′ Σ b n u ^1/2 u.

Theonsistenyof

q b n (α n u)

^{isexaminedin thenextresult.}

Theorem3. Let

u ∈ S ^d−1

^{andassumethat}

α n ↑ 1

^{. If}

E k X k ² < ∞

^then

√ 1 − α n ( b q n (α n u) − q(α n u)) → 0

^{almost surely as}

n → ∞ .

This resultatually means that theextreme geometri quantileestimator is relatively onsistentin the

sensethat

b

q n (α n u) − q(α n u)

k q(α n u) k → 0

^{almostsurelyas}

n → ∞ ,

sine

k q(α n u) k ⁻¹ = O( √

1 − α n )

^{, see Theorem 2(ii). This} normalisation ould be expeted sine the quantity to be estimated diverges in magnitude. Under the additional assumption that

X

^{has a nite}

fourthmoment,anasymptotinormalityresultanbeestablishedforthisestimator:

Theorem4. Let

u ∈ S ^d−1

^{andassumethat}

α n ↑ 1

^issuhthat

n(1 − α n ) → 0

^{. If}

E k X k ⁴ < ∞

^then

p n(1 − α n ) ( q b n (α n u) − q(α n u)) −→ ^d Z

^as

n → ∞

where

Z

^{isaGaussian entredrandom vetor.}

Letus highlight that the ovarianematrix of the Gaussian limit in Theorem 4 essentially depends on

theovarianematrix

M

^{oftheGaussian limitof}

√ n( Σ b n − Σ)

^{, seetheproofin Setion6. Althoughthe}

matrix

M

^{has aheavy and ompliated expression (see e.g. Neudeker and Wesselman, 1990), it an}

beestimated when

E k X k ⁴ < ∞

^{, whihmakesitpossibleto onstrutasymptotiondene regionsfor}

extremegeometri quantiles.

Extremegeometriquantilesanthusbeonsistentlyestimatedby

q b n (α n u)

^{,whatevertheorder}

α n

^,and

anasymptotinormalityresultisobtainedwhen

α n ↑ 1

^{quiklyenough. Theproposedestimatoristhus}

abletoextrapolatearbitrarilyfarfromtheoriginalsample. Thisisverydierentfromtheunivariatease,

where the empirial quantile

b q n (α n ) = inf { t ∈ R | F b (t) ≥ α n }

^{, dedued from the empirial umulative}

distributionfuntion

F b

^{, estimatesthe true quantile}

q(α n )

onsistently onlyif

α n

^{onverges to 1slowly}

enough. Theextrapolationwithfasterrates

α n

^{isthenhandledassumingthattheunderlying}distribution funtion isheavy-tailed andbyusing adapted estimators,seee.g. Weissman(1978)and themonograph

bydeHaanandFerreira(2006).

4 Numerial illustrations

Inthis setion, our main results are illustrated, partiularly Theorems 2, 3and 4in the bivariate ase

d = 2

^{to makethedisplayeasier. Inthis framework,}

u ∈ S ¹

^anberepresentedbyanangle andwemay write

u = u θ = (cos θ, sin θ)

^,

θ ∈ [0, 2π)

^{. The}iso-quantileurves

C q(α) = { q(αu θ ), θ ∈ [0, 2π) }

^{and their}

estimates

Cb q n (α) = {b q n (αu θ ), θ ∈ [0, 2π) }

^{anthenbeonsideredinordertogetagraspofthebehaviour}

of extreme quantiles in everydiretion. The following two distributions are onsidered forthe random

vetor

X

^:

•

^{theentredGaussian}multivariatedistribution

N (0, v X , v Y , v XY )

^,withprobabilitydensityfuntion:

∀ x, y ∈ R, f(x, y) = 1 2π √

det Σ exp



 − 1 2



 x y





′

Σ ⁻¹



 x y









^with

Σ =



 v X v XY

v XY v Y



 .

•

^adouble exponentialdistribution

E (λ − , µ − , λ + , µ + )

^{, with}

λ −

^,

µ −

^,

λ +

^,

µ + > 0

^{, whose} probability

∀ x, y ∈ R, f(x, y) =

 

 



 

  λ + µ +

4 e ^−λ

⁺

^|x|−µ

⁺

^|y|

^if

xy > 0, λ − µ −

4 e ^−λ

⁻

^|x|−µ

⁻

^|y|

^if

xy ≤ 0.

Inthisase,

X

^{isentredandhasovarianematrix}

Σ =



 

 

1 λ ² ₋ + 1

λ ² ₊

1 2

1

λ + µ + − 1 λ − µ −

1 2

1

λ + µ + − 1 λ − µ −

1 µ ² ₋ + 1

µ ² ₊



 

  .

Inourstudy, threedierentsetsof parameterswereused foreah distribution,inorder thattherelated

ovarianematriesoinide:

• N (0, 1/2, 1/2, 0)

^and

E (2, 2, 2, 2)

^{withspherialovarianematries;}

• N (0, 1/8, 3/4, 0)

^and

E (4, 2 p

2/3, 4, 2 p

2/3)

^{withdiagonalovarianematries;}

• N (0, 1/2, 1/2, 1/6)

^and

E (2 √ 3, 2 √

3, 2 p

3/5, 2 p

3/5)

^{withfullovarianematries.}

Ineahase,wearryoutthefollowingomputations:

•

^foreah

α ∈ { 0.99, 0.995, 0.999 }

^{,thetruequantileurves}

C q(α)

^{obtainedbysolvingproblem(1)nu-}

merially,aswellastheiranalogues

C q eq (α)

^usingapproximation(4)areomputed. Thenormalised squaredapproximationerror

e(α) = (1 − α) Z 2π

0 k q eq (αu θ ) − q(αu θ ) k ² dθ

isthenreorded.

•

^{for eah value of}

α

^{, wedraw}

N = 1000

repliationsof an

n −

^sample

(X 1 , . . . , X n )

^of independent opiesof

X

^,with

n ∈ { 100, 200, 500 }

^{. Theestimated quantileurves}

C q b ^(j) n (α)

orrespondingtothe

j −

^{threpliationandtheassoiatednormalisedsquarederror}

E _n ^(j) (α) = (1 − α)

Z 2π 0

b q ^(j) _n (αu θ ) − q(αu θ )

2

dθ

areomputedaswellas themeansquarederror

E n (α) = N ⁻¹ P N

j=1 E ^(j) n (α)

^.

Thetruequantileurves,aswellastheapproximatedandtheestimatedonesaredisplayedonFigures16

in the ase

n = 200

^and

α = 0.995

^{. The true quantile urves look very similar on Figures 1 and 4,}

on Figures 2 and 5 and Figures 3 and 6. This is in aordane with Theorem 2: eventually, extreme

geometriquantilesonlydepend ontheovarianematrixoftheunderlying distribution. Moreover,the

approximatedquantilesurvesare lose tothe trueones in all ases,and the estimated quantile urves

aresatisfyingin allsituationswithamoderatevariability. Similarresultswereobservedfor

n = 100, 500

and

α = 0.99, 0.999

^{. We do not report the graphshere for the sakeof brevity; wedo howeverdisplay}

theapproximationandestimationerrorsinTable1. Unsurprisingly,theestimationerror

E n (α)

^dereases

asthesample size

n

^{inreases. Both} approximationandestimation errors

e(α)

^and

E n (α)

^haveastable

behaviourwithrespetto

α

^.

5 Conluding remarks

In this paper, we established the asymptotis, both in diretion and magnitude, of extreme geometri

quantiles. A partiularonsequeneof ourresultsis thatiftheunderlyingdistribution possessesanite

ovariane matrix

Σ

^{, then an extreme geometri quantile may be estimated aurately, no matter how}

extremeitis,withthehelpofthestandardempirialestimatorof

Σ

^{. Thisissupportedbyournumerial}

results.

This work, however, was arried out under moment onditions suh asthe existene of nite rst and

seond-ordermomentsfor

k X k

^{. It would denitelybe}interestingto seeifouronlusions arryover,to someextent,totheasewhentheseassumptionsareviolated. Furthermore,althoughgeometriquantiles

make an appealing andidate for multivariate quantiles, they lak a ouple of nie properties suh as

aneequivariane, forinstane. Totaklethis issue, onemayapply atransformation-retransformation

proedure,seeSering(2010);suhproedures admitsampleanalogues,see forinstaneChakrabortyet

al. (1998)and Chakraborty(2001). Futureworkon extremegeometri quantilesthus inludes building

andstudyingananalogueofourestimatorfortransformed-retransformeddata.

6 Proofs

SomepreliminaryresultsareolletedinParagraph6.1,theirproofsarepostponedtoParagraph6.3. The

proofsofthemain resultsareprovidedinParagraph6.2.

6.1 Preliminary results

TherstlemmaprovidessometehnialtoolsneessarytoshowTheorem2(ii).

Lemma1. Let

ϕ : R ^d × R + × S ^d−1 → R

^{bethe funtion denedby}

ϕ(x, r, v) = r ²

1 + h x − rv, v i k x − rv k

.

Then forall

v ∈ S ^d−1

^,

ϕ( · , · , v)

^isnonnegativeandwehave that

∀ x ∈ R ^d , ∀ r ≤ k x k , ϕ(x, r, v) ≤ 2r ²

^and

∀ r > k x k , ϕ(x, r, v) ≤ k x k ² .

Inpartiular,

ϕ(x, r, v) ≤ 2 k x k ²

^{for every}

(x, r, v) ∈ R ^d × R + × S ^d−1

^.

Thenextlemmaistherststepto proveTheorem2(i).