ASYMPTOTIC MULTIVARIATE EXPECTILES

Transcription

1 ASYMPTOTIC MULTIVARIATE EXPECTILES Véronique Maume-Deschamps Didier Rullière Khalil Said To cite this version: Véronique Maume-Deschamps Didier Rullière Khalil Said ASYMPTOTIC MULTIVARIATE EX- PECTILES 207 <hal > HAL Id: hal Submitted on 8 Apr 207 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents whether they are published or not The documents may come from teaching and research institutions in France or abroad or from public or private research centers L archive ouverte pluridisciplinaire HAL est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche publiés ou non émanant des établissements d enseignement et de recherche français ou étrangers des laboratoires publics ou privés

2 ASYMPTOTIC MULTIVARIATE EXPECTILES VÉRONIQUE MAUME-DESCHAMPS DIDIER RULLIÈRE AND KHALIL SAID Abstract In [6] a new family of vector-valued ris measures called multivariate expectiles is introduced In this paper we focus on the asymptotic behavior of these measures in a multivariate regular variations context For models with equivalent tails we propose an estimator of these multivariate asymptotic expectiles in the Fréchet attraction domain case with asymptotic independence or in the comonotonic case Introduction In few years expectiles became an important ris measure among more used ones essentially because it satisfies both coherence and elicitability properties In dimension one expectiles are introduced by Newey and Powell 987 [] For a random variable X with finite order 2 moment the expectile of level α is defined as e α X arg min x R E[αX x2 + + αx X 2 +] where x + maxx 0 Expectiles are the only ris measure satisfying both elicitability and coherence properties according to Bellini and Bignozzi 205 [2] In higher dimension one of the proposed extensions of expectiles in [6] is the Matrix Expectiles Consider a random vector X X X d T R d having order 2 moments and let Σ π ij ij d be a d d real matrix symmetric and positive semi-definite such that i { d} π ii π i > 0 A Σ expectile of X is defined as e Σ αx arg min x R d E[αX x T +ΣX x + + αx x T ΣX x ] where x + x + x d + T and x x + We shall concentrate on the case where the above minimization has a unique solution In [6] conditions on Σ ensuring the uniqueness of the argmin are given it is sufficient that π ij 0 i j { d} We shall mae this assumption throughout this paper Then the vector expectile is unique and it is the solution of the following equation system 0 α π i E[X i + {X >x }] α π i E[ X i + {x >X }] { d} i i When π ij for all i j { d} 2 the corresponding Σ-expectile is called a L -expectile It coincides with the L -norm expectile defined in [6] In [6] it is proved that eσ αx X F and α 0 eσ αx X I where X F R {+ } d is the right endpoint vector x F xd F T and by X I R { } d is the left endpoint vector x I xd I T of the support of the random vector X The asymptotic levels ie α or α 0 represent extreme riss For example the solvency thresholds in insurance are generally asymptotic so that the asymptotic behavior of the ris measures is of natural importance The multivariate expectiles can be estimated in the general case using stochastic optimization algorithms The example of estimation by the Robbins-Monro s 95 [2] algorithm presented in [6] shows that in the asymptotic level the obtained estimation is not satisfactory in term of convergence speed This leads us to the theoretical analysis of the asymptotic behavior of multivariate expectiles Date: April Mathematics Subject Classification 62H00 62P05 9B30 Key words and phrases Ris measures multivariate expectiles regular variations extreme values tail dependence functions

3 We shall wor focus on the equivalent tails model It is often used in modeling the claim amounts in insurance in studying dependent extreme events and in ruin theory models This model includes in particular the identically distributed portfolios of riss and the case with scale difference in the distributions In this paper we study the asymptotic behavior of multivariate expectiles in the multivariate regular variations framewor We focus on marginal distributions belonging to Fréchet domain of attraction This domain contains the heavy-tailed distributions that represent the most dangerous claims in insurance Let us remar that the attention to univariate expectiles is recent In [3] asymptotic equivalents of expectiles as a function of the quantile of the same level for regular variation distributions are proved First and second order asymptotics for the expectile of the sum in the case of FGM dependence structure are given in [5] The paper is constructed as follows The first section is devoted to the presentation of the multivariate regularly varying distribution framewor The study of the asymptotic behavior of the multivariate expectiles for Fréchet model with equivalent tails is the subject of Section 2 The case of an asymptotically dominant tail is analyzed in Section 3 Section 4 is a presentation of some estimators of the asymptotic expectiles in the cases of asymptotic independence and comonotonicity The MRV Framewor Regularly varying distributions are well suited to study extreme phenomenons Lots of wors have been devoted to the asymptotic behavior of usual ris measures for this class of distributions and results are given for sums of riss belonging to this family It is well nown that the three domains of attraction of extreme value distributions can be defined using the concept of regular variations see [8; 9; 6; 4] This section is devoted to the classical characterization of multivariate regular variations which will be used in the study of the asymptotic behavior of multivariate expectiles We also recall some basic results on the univariate setting that we shall use Univariate regular variations We begin by recalling basic definitions and results on univariate regular variations Definition Regularly varying functions A measurable positive function f is regularly varying of index ρ at a {0 + } if for all t > 0 ftx x a fx tρ we denote f RV ρ a A slowly varying function is a regularly varying function of index ρ 0 Remar that a function f RV ρ + if and only if there exists a slowly varying function at infinity L RV 0 + such that fx x ρ Lx Theorem 2 Karamata s representation [20] For any slowly varying function L at + there exist a positive measurable function c that satisfies cx c ]0 + [ and a measurable function ε with εx 0 such that x + x Lx cx exp εt dt t x + The Karamata s representation is generalized to RV functions Indeed f RV ρ + if and only if it can written in the form x ρt fx cx dt t where ρt t ρ and ct t c ]0 + [ Throughout the paper we shall consider generalized inverses of nondecreasing functions f: f y inf{x R fx y} Lemma 3 Inverse of RV functions [4] Let f RV ρ a be non decreasing If ρ > 0 then f RV /ρ fa and if ρ < 0 then f RV /ρ fa In [9] it is proven that for a measurable non negative function f defined on R + such that f RV ρ + if and only if f RV ρ + 2 fx + x +

4 Lemma 4 Integration of RV functions Karamata s Theorem [8] For a positive measurable function f regularly varying of index ρ at + locally bounded on [x 0 + with x 0 0 if ρ > then if ρ < then x + x + x x x 0 ftdt xfx xfx ftdt ρ + ρ + Lemma 5 Potter s bounds [4] For f RV ρ a with a {0 } et ρ R For any 0 < ɛ < and all x and y sufficiently close to a we have x ρ ɛ ρ+ɛ x x ɛ min fx ρ ɛ ρ+ɛ x + ɛ max y y fy y y Many other properties of regularly varying functions are presented eg in [4] 2 Multivariate regular variations The multivariate extension of regular variations is introduced in literature v in [7] We denote by µ n µ the vague convergence of Radon measures as presented in [2] The following definitions are given for non negative random variables Definition 6 Multivariate regular variations The distribution of a random vector X on [0 ] d is said to regularly varying if there exist a non-null Radon measure µ X on the Borel σ-algebra B d on [0 ] d \0 and a normalization function b : R R which satisfies bx + such that x + X up bu υ µ X as u + There exist several equivalent definitions of multivariate regular variations which will be useful in what follows Definition 7 MRV equivalent Definitions Let X be a random vector on R d the following definitions are equivalent: The vector X has a regularly varying tail of index θ There exist a finite measure µ on the unit sphere S d and a normalization function b : 0 0 such that 2 t + P X > xbt X X x θ µ for all x > 0 The measure µ depends on the chosen norm it is called the spectral measure of X There exists a finite measure µ on the unit sphere S d a slowly varying function L and a positive real θ > 0 such that 3 x + for all B BS d with µ B 0 x θ Lx P X > x X X B µb From now on MRV denotes the set of multivariate regularly varying distributions and by MRVθ µ denotes the set of random vectors with regularly varying tail with index θ and spectral measure µ From 3 we may assume that µ is normalized ie µs d which implies that X has a regularly varying tail of index θ On another hand X P x + X B X > x x + x + X P X > x X B x θ P X > x Lx µbx θ Lx µb for all B BS d with µ B 0 That means that conditionally to { X > x} The different possible characterizations of the MRV concept are presented in [8] 3 X X converges wealy to µ

5 3 Characterization using tail dependence functions Let X X X d be a random vector From now on F Xi denotes the survival function of X i We assume that X has equivalent regularly varying marginal tails which means: H: FX RV θ + with θ > 0 H2: The tails of X i i d are equivalent That is for all i {2 d} there is a positive constant c i such that F Xi x x + F X x c i H and H2 imply that all marginal tails are regularly varying of index θ at + In this paper we use the definition of the upper tail dependence function as introduced in [3] Definition 8 The tail dependence function Let X be a random vector on R d with continuous marginal distributions The tail dependence function is defined by 4 λ X U x x d t 0 t P F X X tx F Xd X d tx d when the it exists For d denote by X a dimensional sub-vector of X C its copula and C its survival copula The upper tail dependence function is 5 λ C tu tu U u u t 0 + t if this it exists The lower tail dependence function can be defined analogically by λ C tu tu Lu u t 0 + t when the it exists In this paper our study is ited to the upper version as defined in 5 The following two theorems show that under H and H2 the MRV character of multivariate distributions is equivalent to the existence of the tail dependence functions Theorem 9 Theorem 23 in [4] Let X X X d be a random vector in R d with continuous marginal distributions F Xi i d that satisfy H and H2 If X has an MRV distribution the tail dependence function exists and it is given by par /θ /θ λ U u u xp u ud X > bx X > bx x + for any { d} Theorem 0 Theorem 32 in [23] Let X X X d be a random vector in R d with continuous marginal distributions F Xi i d that satisfies H and H2 If the tail dependence function λ U exists for all { d} then X is MRV its normalization function is given by bu F u and the spectral measure is X µ[0 x] c i c i x θ i i<j d λ 2 U c i x θ i c c j x θ j + + d+ λ d U c x θ c dx θ d By construction of the multivariate expectiles only the bivariate dependence structures are taen into account We shall use the functions λ XiX U for all i { d} 2 In order to simplify the notation we denote it by λ i U If the vector X has an MRV distribution the pairs X i X j have also MRV distributions for any i j { d} 2 So in the MRV framewor and under H and H2 the existence of functions λ i is insured In addition we assume in all the rest of this paper that these functions are continuous 2 Fréchet model with equivalent tails In this section we assume that X satisfies H and H2 with θ > It implies that X belongs to the extreme value domain of attraction of Fréchet MDAΦ θ This domain contains distributions with infinite endpoint x F sup{x : F x < } + so as α we get e i αx + i Also from Karamata s Theorem Theorem 4 we have for i d E[X i x + ] 2 x + x F Xi x θ 4 c d

6 for all i { d} Proposition 2 Let Σ π ij ijd with π ij > 0 for all i j { d} Under H and H2 the components of the multivariate Σ-expectiles e α X e i αx id satisfy e i 0 < αx e αx e i αx e < + i {2 d} αx Proposition 2 implies that distributions with equivalent tails have asymptotically comparable multivariate expectile components Before we prove Proposition 2 we shall demonstrate some preinary results Firstly let X X X d T satisfy H and H2 we denote e i αx for all i { d} We define the functions lx α ix j for all i j { d} 2 by 22 l α X ix j x j αe[x i + {Xj>x j}] αe[x i {Xj<x j}] and l α X i l α X ix i The optimality system 0 rewrites 23 l α X x We shall use the fellowing sets: ii π i π l α X ix x { d} J i 0 {j { d} \ {i} J i C {j { d} \ {i} 0 < x j x j 0} < x j x j < + } and J i {j { d} \ {i} + } The proof of Proposition 2 is written for π ij for all i j { d} 2 ie for the L -expectiles The general case can be treated in the same way provided that π ij > 0 for all i j { d} 2 The proof of Proposition 2 follows from Lemma 22 and Propositions 23 and 24 below Lemma 22 Assume that H and H2 are satisfied If t os then for all i j { d} 2 t + s F Xi s t F Xj t 0 2 If t Θs then for all i j { d} 2 F Xi s F Xj t c i s θ as t c j t Proof We give some details on the proof for the first item the second one may be obtained in the same way Under H and H2 for all i { d} F Xi RV θ + there exists for all i a positive measurable function L i RV 0 + such that F Xi x x θ L i x x > 0 then for all i j { 2} 2 and all t s > 0 24 and under H2 s F Xi s t F Xj t s t 25 θ+ Li s s θ+ L j t Li s t L i t x + L i x L j x c i c j L i t L j t Using Karamata s representation for slowly varying functions Theorem 2 there exist a constant c > 0 a positive measurable function c with cx c > 0 such that ɛ > 0 t 0 such that t > t 0 x + Recall that t Θs means that there exist positive constants C and C 2 such that C s t C 2 s 5

7 Taing 0 < ɛ < θ we conclude t + L i s s ɛ cs L i t t ct s F Xi s t F Xj t 0 i j { d}2 Proposition 23 Under H and H2 the components of the asymptotic multivariate expectile satisfy Proof Using H2 it is sufficient to show that α F X e α α < + i {2 d} F Xi e i αx Assume that a subsequence α n we may assume that We have l α X x αx α F X e αx < + + we shall prove that in that case 23 cannot be satisfied Taing if necessary X α + F X e α X αe[x x + ] αe[x x ] αx 2α E[X x + ] αx E[X ] αx αx F X x α + E[X ] x 2α E[X x + ] x FX x Furthermore for all i {2 d} lx α ix x αe[xi + {X>x }] αe[x i {X<x }] αx On one side α recall 2 αx αe[xi + {X>x }] E[X i + {X<x }] PX < x + E[X i {X<x }] αx x x x E[X i + {X>x }] αx E[X i + ] So that Lemma 22 implies F X x αx α E[X i + ] FXi E[X i + {X>x 26 }] 0 JC J αx Let i J0 taing if necessary a subsequence we may assume that xi x 0 27 Now Thus E[X i + {X>x }] αx x x P X i > t X > x dt αx 28 P X i > t X > x dt x αx + x x FXi x FX x i {2 d} P X i > t X > x dt αx P X > x dt F X x x i αx α x P X i > t X > x dt αx 0 6

8 Consider the second term of 27 Karamata s Theorem Theorem 4 gives which leads to x P X i > t X > x dt 29 Finally we get x αx x P X > t dt αx α θ x P X i > t X > x dt P X i > t dt αx F X x α αx 0 E[X i + {X>x 20 }] 0 i J0 αx We have shown that E[X x + {X>x }] 0 {2 d} αx so the first equation of optimality system 23 implies that α J 0 \J l α X X x x αx + J C l α X X x x αx + this is absurd since the x s are non negative and consequently J α F X x < + lx α X x x αx 2 Proposition 24 Under H and H2 the components of the asymptotic multivariate expectile satisfy α 0 < i {2 d} F Xi e i αx Proof Using H2 it is sufficient to show that α F X e α Let us assume that a convergent subsequence we may assume that l α X x x FX x α 0 < F X e αx x x X 0 we shall see that in that case 23 cannot be satisfied Taing if necessary α F X e α 0 In this case X 2α E[X x + ] α x FX x F x E[X ] x α θ > 0 On another side let i J taing if necessary a subsequence we may assume that x o Lemma 22 and Proposition 23 give: α α F x x F Xi xif Xi 0 as α x F X x Moreover E[X i + {X>x }] E[X i + ] E[X i + ] FXi x FX x x FX x FXi x FX x We deduce lx α ix x αe[xi + {X>x }] αe[x i {X<x }] x FX x x FX x 0 i J 7

9 Going through the it α in the fist equation of the optimality system 23 divided by x FX x leads to lx α X x x x FX x θ which is absurd because lx α X x x x FX x J 0 J C \J We can finally conclude that J 0 J C \J J0 J C \J J0 J C \J J0 J C \J The combination of Propositions 23 and 24 gives 0 < We may now prove Proposition 2 α F Xi e i αx αe[x x + {X>x }] αe[x x {X<x }] x FX x E[X x + {X>x }] x FX x E[X x + {X>x }] 0 x FX x α F X x > 0 α F X x α < + i {2 d} F Xi e i αx Proof of Proposition 2 We shall prove that J the fact that J for all { d} may be proven in the same way This implies that J 0 J for all { d} hence the result We suppose that J let i J taing if necessary a subsequence we may assume that /x + as α From Propositions 23 and 24 we have 0 < α F Xi e i αx α < + i {2 d} F Xi e i αx so taing if necessary a subsequence we may assume that l R \{+ } such that In this case Moreover We get α F X x l x x lx α x x FX x 2α E[X x + ] α x FX x F x E[X ] x θ l < + E[X i + {X>x }] x FX x lx α ix x x FX x E[X i + ] x FX x E[X i + ] FXi FXi 0 using Lemme 22 x FX x αe[xi + {X>x }] αe[x i {X<x }] x FX x E[Xi + {X>x }] α x FX x F X x x i J Going through the it α in the first equation of the optimality System 23 divided by x FX x leads to lx α 2 X x x x FX x J 0 J C \J 8

10 Now let J 0 E[X x + {X>x }] x FX x x x x Karamata s Theorem Theorem 4 leads to Consider J C and Finally we deduce that J 0 J C \J x x x P X > x dt + x FX x E[X x + {X>x }] x FX x x P X > t X > x dt + x FX x P X > x dt + x FX x x x P X > t dt x FX x E[X x + ] x FX x E[X x + ] x FX x FX x x FX x x J0 J C \J J0 J C \J J C θ x P X > t dt x FX x P X > t X > x dt x FX x + θ J 0 E[X x + ] x FX x θ l α X X x x x FX x l α X X x x x x FX x x x θ+ l x x θ+ x FX x x FX x + J 0 \J + c θ This is contradictory with 2 and consequently J is necessarily an empty set The result follows Proposition 25 Asymptotic multivariate expectile Assume that H and H2 are satisfied and X has a regularly varying multivariate distribution in the sense of Definition 6 consider the L -expectiles e α X e i αx id α Then any it vector η β 2 β d of F X e X e2 α X e α X ed α X α e α X satisfies the following equation system 22 θ η β θ λ i ci c U t θ dt η βθ { d} ii β By solving the system 22 we may obtain an equivalent of the asymptotic multivariate expectile using the marginal quantiles Proof The optimality system 23 can be written in the following form 2α E[X x + ] x FX x α E[X ] F X x x ii ii For all { d} we have taing if necessary a subsequence 2α E[X x + ] x FX x α F X x 9 α E[X i {X <x }] x FX x α E[X i + {X >x }] { d} x FX x E[X ] x θ η β θ

11 and for all i { d} \ {} αe[x i {X <x }] α xi PX i < X < x E[X i {Xi<X <x }] x FX x F X x x x Moreover Firstly we remar that Since the functions λ i U η βθ η βθ β E[X i + {X >x }] x FX x x FX x x β x + x PX i > t X > x dt PX i > tx X > x dt F X x P FXi X i < F Xi tx F X X < F X x F X x P FXi X i < F Xi tx F X X < F X x dt F X x x β are assumed to be continuous P FXi X i < 23 F Xi tx F X X < F X x λ F i U X x In order to show that αe[x i + {X >x }] x FX x β ci t θ λ i ci U t θ dt we may use the Lebesgue s Dominated Convergence Theorem with Potter s bounds 942 Lemma 5 for regularly varying functions First of all P FXi X i < F Xi tx F X X < F X x { min F } Xi tx F X x F X x since F Xi tx F F Xi tx F X tx F and Xi tx X x F X tx F X x F ci X tx using Potter s bounds for all ε > 0 and 0 < ε 2 < θ there exists x 0 ε 2 ε such that for min{x tx } x 0 ε 2 ε Lebesgue s theorem gives x so for all i { d} 2 F Xi tx F X x ci + 2ε t θ maxt ε2 t ε2 P FXi X i < F Xi tx F X X < F X x F X x E[X i + {X >x }] x FX x Hence the system announced in this proposition β dt β λ i ci U t θ dt dt λ i ci U t θ dt In the general case of Σ-expectiles with Σ π ij ijd π ij 0 π ii π i > 0 System 22 becomes θ η β θ π + i λ i ci π U t θ dt η βθ { d} ii β 0

12 Moreover let us remar that System 22 is equivalent to the following system 24 i β λ i U ci t θ β θ dt i λ i U ci t θ dt {2 d} The it points are thus completely determined by the asymptotic bivariate dependencies between the marginal components of the vector X Proposition 26 Assume that H and H2 are satisfied and the multivariate distribution of X is regularly varying in the sense of Definition 6 consider the L -expectiles e α X e i αx id Then any it vector η β 2 β d α of F X e X e2 α X e α X ed α X α e α X satisfies the following system of equations { d} 25 θ η β θ c + θ+ i βi λ i U t θ c θ β dt η βθ β c i ii Proof The proof is straightforward using a substitution in System 22 and the positive homogeneity property of the bivariate tail dependence functions λ i U see Proposition 22 in [0] The main utility of writing the asymptotic optimality system in the form 25 is the possibility to give an explicit form to η β 2 β d for some dependence structures Example: Consider that the dependence structure of X is given by an Archimedean copula with generator ψ The survival copula is given by Cx x d ψψ x + + ψ x d where ψ x inf{t 0 ψt x} see eg [7] for more details Assume that ψ is a regularly varying function of a non-positive index ψ RV θψ According to [5] the right tail dependence functions exist and one can get their explicit forms θψ λ U x x x θ ψ i Thus the bivariate upper tail dependence functions are given by λ i U t θ c θ β t θ θ ci θ ψ ψ + c i i β θ θ ψ θψ In particular if θ θ ψ we have λ i U and System 25 becomes θ η β θ t θ c i β ii θ θ dt + θ c i β + ci ci θ β θ+ θ θ+ η βθ Lemma 27 The comonotonic Fréchet case Under H and H2 consider the L -expectiles e α X e i αx id If X X X d is a comonotonic random vector then the it α η β 2 β d F X e αx e2 αx e αx ed αx e αx satisfies α F X e αx θ and β c /θ { d}

13 Proof Since the random vector X is comonotonic its survival copula is We deduce the expression of the functions λ ij U So i λ i U i C X u u d minu u d u u d [0 ] d λ ij U x j min x j x j R 2 + i j { d} t θ β θ dt c i ii β min t θ c i θ c c i + β + θ β + c c i c i θ dt θ β θ + θ+ Under assumptions H and H2 and by Proposition 25 let η β 2 β d be a solution of the following equation system η c i β θ+ i β θ θ β β c θ i c i + θ ii c i β θ+ i + β + c c i θ + θ+ { d} η θ and β c θ is the only solution to this system Proposition 28 Asymptotic independence case Under H and H2 consider the L -expectiles e α X e i αx id If X X X d is such that the pairs X i X j are asymptotically independent then the it vector η β 2 β d α of F X e X e2 α X e α X ed α X α e α X satisfies for all { d} η θ + c θ j j2 Proof The hypothesis of asymptotic bivariate independence means: and β c θ PX i > X j > x j PX i > tx j X j > x j 0 PX j > x j PX j > x j for all i j { d} 2 and for all t > 0 then Lebesgue s Theorem used as in Proposition 25 gives E[X i + {Xj>x j}] x j FXjx j 0 The asymptotic multivariates expectile verifies the following equation system which can be rewritten as 26 θ η β θ + ii ηθ β θ x j PX i > tx j X j > x j dt PX j > x j η β θ { d} i 2 { d}

14 hence β c θ for all { d} and η θ + In the general case of a matrix of positive coefficients π ij same but the it η will change: for all { d} We remar that e αx e αx c θ c θ j j2 α and F X e αx c θ + i j { d} the its i 2 d remain the θ π j c θ j π α F Xi c i θ which allows a comparison between the marginal quantile and the corresponding component of the multivariate expectile and since F X is regularity varying function at 0 for all { d} with index θ see Lemma 3 we get + c θ θ i e αx VaR α X θ θ i2 where VaR α X denotes the Value at Ris of X at level α ie the α-quantile F X α of X These conclusions coincide with the results obtained in dimension for distributions that belong to the domain of attraction of Fréchet in [3] The values of constants c i determine the position of the marginal quantile compared to the corresponding component of the multivariate expectile for each ris θ c θ 3 Fréchet model with a dominant tail This section is devoted to the case where X has a dominant tail with respect to the X i s We shall not give the proofs they are an adaptation to the proofs of Section 2 We shall only state the intermediate needed results and leave the proofs to the reader Proposition 3 Asymptotic dominance Under H consider the L -expectiles e α X e i αx id If then and F Xi x x + F X x 0 α e i αx e αx 0 j2 i {2 d} dominant tail hypothesis α α α F Xi e i αx 0 α F X e αx θ The proof of Proposition 3 follows from the following three lemmas Lemma 32 Under H consider the L -expectiles e α X e i αx id If then F Xi x x + F X x 0 i {2 d} i {2 d} E[X i + {X>x }] 0 i {2 d} α x FX x 3

15 Lemma 33 Under H consider the L -expectiles e α X e i αx id If then F Xi x x + F X x 0 α i {2 d} α F X e αx < + Lemma 34 Under H consider the L -expectiles e α X e i αx id If then F Xi x x + F X x 0 i {2 d} α 0 < α F X e αx Proposition 3 shows that the dominant ris behaves asymptotically as in the univariate case and its component in the asymptotic multivariate expectile satisfies e αx α θ θ VaRα X α e α X the right equivalence is proved in the univariate case in [3] Proposition 23 Example: Consider Pareto distributions X i P aa i b i d such that a i > a for all i { d} The tail of X dominates that of the X i s and Proposition 3 applies 4 Estimation of the asymptotic expectiles In this section we propose some estimators of the asymptotic multivariate expectile We focus on the cases of asymptotic independence and comonotonicity for which the equation system is more tractable We begin with the main ideas of our approach then we construct the estimators using the extreme values statistical tools and prove its consistency We terminate this section with a simulation study Proposition 4 Estimation s idea Using notations of previous sections consider the L -expectiles e α X e i α αx id Under H H2 and the assumption that the vector F X e X e2 α X e α X ed α X α e α X has a unique it point η β 2 β d e α X T VaR α X η θ β2 β d T α Proof Let η β 2 β d α F X e X e2 α X e α X ed α X α e α X we have Moreover α F X e α X index θ This leads to and the result follows e α X T e αx β 2 β d T η and Theorem 52 in [4] states that F X is regularly varying at 0 with e αx FX θ α η Proposition 4 gives a way to estimate the asymptotic multivariate expectile Let X X X d T be an independent sample of size n of X with X i X i X di T for all i { n} We denote by X in X i2n X inn the ordered sample corresponding to X i 4 Estimator s construction We begin with the case of asymptotic independence Propositions 28 and 4 are the ey tools in the construction of the estimator We have for all i { d} c θ i and α F Xi e i αx 4 c θ i θ + c θ j j2

16 Proposition 4 gives e α X α VaR α X θ θ + i2 c θ θ i c θ 2 c So in order to estimate the asymptotic multivariate expectile we need an estimator of the univariate quantile of X of the tail equivalence parameters and of θ In the same way and for the case of comonotonic riss we may use Proposition 27 and by Proposition 4 we obtain α F Xi e i αx θ and c /θ i i { d} e α X T α VaR α X θ θ c θ 2 c θ d T The X i s have all the same index θ of regular variation which is also the same as the index of regular variation of X We propose to estimate θ by using the Hill estimator γ We shall denote θ See [9] for details on the Hill γ estimator In order to estimate the c i s we shall use the GPD approximation: for u a large threshold and x u F x F x θ u u Let N be fixed and consider the thresholds u i : θ d T F Xi u i F X u n i { d} The u i are estimated by X in +n with and /n 0 as n Using Lemma 22 we get θ ui c i n We shall consider 4 ĉ i u Xin +n X n +n ˆγ where ˆγ is the Hill s estimator of the extreme values index constructed using the largest observations of X Let θ γ Proposition 42 Let n be such that and /n 0 as n Under H and H2 for any i 2 d ĉ i P c i Proof The results in [] page 86 imply that for any i d X in +n u i P Moreover this is well nown see [9] that the Hill estimator is consistent Using 4 and the fact that we get the result X in +n X n +n u i u in probability and thus is bounded in probability To estimate the extreme quantile we may also use Weissman s estimator 978 [22]: ˆγ n VaR α X X n n+n αn The properties of Weissman s estimator are presented in Embrechts et al 997 [8] and also in [] page 9 Now we can deduce some estimators of the extreme multivariate expectile using the previous ones in the cases of asymptotic independence and perfect dependence 5

17 Definition 43 Multivariate expectile estimator Asymptotic independence Under H and H2 in the case of bivariate asymptotic independence of the random vector X X X d T we define the estimator of the L expectile as follows n ê α X X n n+n αn ˆγ ˆγ ˆγ ˆγ + d ˆγ l2 ĉl ˆγ ˆγ ˆγ ˆγ ĉ2 ĉ Propositions 28 and 42 as well as convergence properties of VaRα X imply that the term by term ratio ê α X/e α X goes to in probability More wor is required to get the asymptotic normality Definition 44 Multivariate expectile estimator comonotonic riss Under the assumptions of the Fréchet model with equivalent tails for a comonotonic random vector X X X d T we define the estimator of L expectile as follows n ê + α X X n n+n αn ˆγ ˆγ ˆγ ˆγ ĉˆγ 2 ĉˆγ d T As above Propositions 28 and 42 as well as convergence properties of VaR α X imply that the term by term ratio ê + α X/e α X goes to in probability 42 Numerical illustration The attraction domain of Fréchet contains the usual distributions of Pareto Student and Cauchy In order to illustrate the convergence of the proposed estimators we consider a bivariate Pareto model X i P aa b i i { 2} Both distributions have the same scale parameter a so they have equivalent tails with equivalence parameter c 2 F X2 x x + F X x x + a b 2 b 2+x In what follows we consider two models for which the exact values of the L -expectiles are computable In the first model the X i s are independent In the second one the X i s are comonotonic In the simulations below we have taen the same n to get ˆγ and ê α X For X i P a2 5 i + i2 and n Figure illustrates the convergence of estimator ĉ 2 when + b b +x a b2 b a ˆγ ˆγ d T Figure Convergence of ĉ 2 X i P a2 5 i+ i2 The left figure concerns the independence model The right one concerns the comonotonicity case In the independence case which is a special case of asymptotic independence the functions l α X ix j defined in 22 have the following expression l α X ix j x j α FXj x j E [ X i + ] α FXj x j E [ X i ] 6

18 In the comonotonic case we have ] lx α ix j x j α FXj x j µ ij + + E [X i max µ ij + α F Xj x j µ ij + + E [X i min µ ij ] where µ ij FX i F Xj x j For Pareto distributions µ ij bi b j x j From these expressions the exact value of the asymptotic multivariate expectile is obtained using numerical optimization and we can confront it to the estimated values Figure 2 presents the obtained results for different n values in the independence case the comonotonic case is illustrated in Figure 3 The simulations parameters are a 2 b 0 b 2 5 and n Figure 2 Convergence of ê α X asymptotic independence case On the left the first coordinate of e α X and ê α X for various values of n are plotted The right figure concerns the second coordinate 7

19 Figure 3 Convergence of ê + α X comonotonic case On the left the first coordinate of e α X and ê α X for various values of n are plotted The right figure concerns the second coordinate Conclusion We have studied properties of extreme expectiles in a regular variations framewor We have seen shows that the asymptotic behavior of expectiles vectors strongly depends on the marginal tails behavior and on the nature of the asymptotic dependence The main conclusion of this analysis is that the equivalence of marginal tails leads to equivalence of the asymptotic expectile components The statistical estimation of the integrals of the tail dependence functions would allow to construct estimators of the asymptotic expectile vectors This paper s estimations are ited to the cases of asymptotic independence and comonotonicity which do not require the estimation of the tail dependence functions The asymptotic normality of the estimators proposed in the last section of this paper requires a careful technical analysis which is not considered in this paper A natural perspective of this wor is to study the asymptotic behavior of Σ-expectiles in the case of equivalent tails of marginal distributions in the domains of attraction of Weibull and Gumbel The Gumbel s domain contains most of the usual distributions especially the family of Weibull tail-distributions which maes the analysis of its case an interesting tas References [] J Beirlant Y Goegebeur J Segers and J Teugels Statistics of extremes: theory and applications John Wiley & Sons 2006 [2] F Bellini and V Bignozzi On elicitable ris measures Quantitative Finance 55: [3] F Bellini and E Di Bernardino Ris management with expectiles The European Journal of Finance 236: [4] N Bingham C M Goldie and J Teugels Regular variation volume 27 Cambridge university press 989 [5] A Charpentier and J Segers Tails of multivariate archimedean copulas Journal of Multivariate Analysis 007: [6] L De Haan and A Ferreira Extreme value theory: an introduction Springer Science & Business Media 2007 [7] L De Haan and S Resnic Limit theory for multivariate sample extremes Zeitschrift für Wahrscheinlicheitstheorie und verwandte Gebiete 404: [8] P Embrechts C Klüppelberg and T Miosch Modelling extremal events volume 33 Springer Science & Business Media 997 8

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