Inference for heavy tailed data applications in insurance and finance 1st edition liang peng All Cha
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Exact Statistical Inference for Categorical Data 1st Edition Shan
Let Ω beaspace,whichisanarbitrary,nonemptyset.Write ω ∈ Ω if ω is anelementof Ω ,andwrite A ⊆ Ω if A isasubsetof Ω .
Definition1.1. Anonemptyclass A ofsubsetsof Ω iscalledan algebra if i) thecomplementaryset Ac ∈ A whenever A ∈ A;and ii) theunion A1 ∪ A2 ∈ A whenever A1 ∈ A and A2 ∈ A. Moreover, A iscalleda σ -algebra ora σ -field if,inadditiontoi)andii), iii) ∪∞ i=1 Ai ∈ A whenever Ai ∈ A for i ≥ 1.
Definition1.2. If A isa σ -algebrawithrespecttothespace Ω ,then thepair (Ω, A) iscalleda measurablespace.Thesetsof A arecalled measurablesets.
Definition1.3. Theelementsofthe σ -algebra B generatedbytheclass ofinfiniteintervalsoftheform [−∞, x) for −∞ < x < ∞ arecalled Borel sets.Themeasurablespace (R =[−∞, ∞], B ) iscalled Borelspace
Definition1.4. If (Ω1 , A1 ) and (Ω2 , A2 ) aretwomeasurablespacesand f isamappingfrom Ω1 to Ω2 ,then f issaidtobea measurabletransformation/mapping if f 1 (A) ∈ A1 forany A ∈ A2 ,where f 1 (A) ={ω : ω ∈ Ω1 , f (ω) ∈ A} .
Definition1.5. Forameasurablespace (Ω, A) ,asetfunction P defined on A iscalleda probability if i) P (∅) = 0,where ∅ denotestheemptyset; ii) P (A ∪ B) = P (A) + P (B) fordisjointevents A, B ∈ A (i.e., A ∩ B =∅ ); iii) P (∪∞ i=1 Ai ) = ∞ i=1 P (Ai ) fordisjointevents Ai ∈ A, i = 1, 2, ··· . Inthiscase, (Ω, A, P ) iscalleda probabilityspace
Definition1.6. Areal-valuedmeasurablefunction X onaprobability space (Ω, A, P ) iscalleda randomvariable.Thefunction
F (x) = P (X ≤ x) := P ({ω ∈ Ω : X (ω) ≤ x}) for x ∈ R = (−∞, ∞)
iscalledthe cumulativedistributionfunction or distributionfunction of X .
Definition1.7. Asequenceofrandomvariables {Xn }∞ n=1 definedona probabilityspace (Ω, A, P ) iscalled independent ifforany m ≥ 1,1 ≤ i1 < ··· < im < ∞ and −∞ < x1 , ··· , xm < ∞ P (Xi1 ≤ x1 , , Xim ≤ xm ) = m j =1 P (Xij ≤ xj ).
Definition1.8. If {Xn }∞ n=0 isasequenceofrandomvariablesonaprobabilityspace (Ω, A, P ) ,then {Xn }∞ n=1 issaidto convergeinprobability to X0 (notation: Xn p → X0 )ifforany > 0 lim n→∞ P (|Xn X0 | > ) = 0.
Definition1.9. If {Xn }∞ n=0 isasequenceofrandomvariablesonaprobabilityspace (Ω, A, P ) withcorrespondingcumulativedistributionfunctions {Fn (x)}∞ n=0 ,then {Xn }∞ n=1 issaidto convergeindistribution to X0 (notation: Xn d → X0 or Xn d → F0 )ifforanycontinuitypoint x of F0
lim n→∞ Fn (x) = F0 (x).
Definition1.10. Asequenceofrandomvariables {Xn } onaprobability space (Ω, A, P ) issaidtobe boundedinprobability ifforany > 0, thereexistconstants C > 0andinteger N suchthat
P (|Xn | > C ) ≤ forall n ≥ N .
Let {Xn } beasequenceofrandomvariablesonaprobabilityspace (Ω, A, P ) and {bn } beasequenceofpositiveconstants.Wewrite Xn = op (bn ) if Xn /bn p → 0,andwrite Xn = Op (bn ) if Xn /bn isboundedinprobability.
Definition1.11. A stochasticprocess isacollection {Xt : t ∈ T } ,where T isasubsetof R and Xt isarandomvariableonaprobabilityspace (Ω, A, P ) .
Definition1.12. A Wienerprocess {W (t ) : t ≥ 0} isacontinuous-time stochasticprocesssatisfying i) W (0) = 0;
ii) W (t + u) W (t ) isindependentofthe σ -algebrageneratedby {W (s) : 0 < s ≤ t } forany u > 0;
iii) W (t + u) W (t ) hasanormaldistributionwithmeanzeroandvariance u forany u > 0.
Definition1.13. If W (t ) for t ≥ 0isaWienerprocess,then B(t ) = W (t ) t T W (T ) iscalleda BrownianBridge on [0, T ] .Inthiscase,
B(0) = B(T ) = 0and E B(s)B(t ) = s(T t ) for0 ≤ s < t ≤ T ,buttheincrementsarenolongerindependent.
Forthespace (E ,ε) ,let CK (E ) bethesetofallcontinuous,realvalued functionson E withcompactsupport,and C + K (E ) bethesubsetof CK (E ) consistingofcontinuous,nonnegativefunctionswithcompactsupport.Let M+ (E ) bethesetofallnonnegativeRadonmeasureson (E ,ε) anddefine μ+ (E ) tobethesmallest σ -fieldofsubsetsof M+ (E ) makingthemaps m → m(f ) = E fdm from M+ (E ) → R measurableforall f ∈ C + K (E ) .Here Radon meansthemeasureofcompactsetsisalwaysfinite.
Definition1.15. ξ isa randommeasure ifitisameasurablemapfrom aprobabilityspace (Ω, A, P ) into (M+ (E ),μ+ (E )) .
Definition1.16. For μn ,μ ∈ M+ (E ) ,wesay μn convergesvaguely to μ (written μn v → μ )if μn (f ) → μ(f ) forall f ∈ C + K (E )
Definition1.17. C ⊂ Rd isa cone if t x ∈ C forevery t > 0and x ∈ C .
Let Λ denotetheclassofstrictlyincreasing,continuousmappingsof [0, 1] ontoitselfwith λ(0) = 0and λ(1) = 1foreach λ ∈ Λ .Given x and y inthespace D[0, 1] ,define d (x, y) tobetheinfimumofthosepositive forwhichthereexistsa λ ∈ Λ suchthat
sup t |λ(t ) t |≤ and sup t |x(t ) y(λ(t ))|≤ .
Inthisway, d (x, y) definesthe Skorohodtopology
Definition1.18. Let F bethecumulativedistributionfunctionofarandomvariable X .Thenthe generalizedinverse of F isdefinedas
F (u) = inf{t : F (t ) ≥ u} for0 < u < 1. (1.1)
Lemma1.1. LetFbeacumulativedistributionfunction.
i) Foranyx ∈ R andu ∈ (0, 1) ,F (u) ≤ xifandonlyifu ≤ F (x) .
ForanyBrownianbridge {B(s) : 0 ≤ s ≤ 1} ,andwith0 ≤ a < b ≤ 1and thefunctions f and g asabovewedefinethefollowingstochasticintegral
andthesameformulafor g replacing f .
1.2BASICEXTREMEVALUETHEORY
Let X1 , ··· , Xn bearandomsampleofsize n fromadistributionfunction F ,thatis, X1 , ··· , Xn areindependentandidenticallydistributed(i.i.d.) randomvariableswithdistributionfunction F .The univariateextreme valuetheory isbasedontheassumptionthatthereexistconstants an > 0 and bn ∈ R suchthat
where G isanon-degeneratedistributionfunction.Inthiscase G iscalled an extremevaluedistribution and F issaidtobeinthe domainof (maximum)attraction oftheextremevaluedistribution G (notation: F ∈ D(G ) ).Toclassify G ,weneedthefollowingdefinition.
Definition1.19. Twodistributionfunctions F (x) and G (x) aresaidto havethe sametype ifforsomeconstants a > 0and b ∈ R G (x) = F (ax + b) forall x ∈ R.
Lemma1.2. Let {Xn }, U , VberandomvariablessuchthatneitherUnorV isdegenerate(i.e.,bothUandVarenon-constant).Ifthereareconstantsan > 0, αn > 0,bn ∈ R, βn ∈ R suchthat
then
Proof. SeeProposition0.2ofResnick [91].
Usingtheabovenotionofthesametype,itiswell-knownthatthe limitingdistribution G in (1.5) mustbeoneofthefollowingthreetypes: • ReversedWeibulldistribution Φα (x) = exp( ( x)α ), x <
• Gumbeldistribution
Λ(x) = exp( e x ), x ∈ R;
• Fréchetdistribution
Ψα (x) = 0, x ≤ 0, exp( x α ), x > 0,
where α> 0.
Aunifiedexpressionfor G in (1.5) is
Here γ ∈ R iscalledthe extremevalueindex,and γ< 0,γ = 0,γ> 0 correspondtothereversedWeibulldistribution,Gumbeldistribution, Fréchetdistribution,respectively.Formodelinglossesininsuranceandfinance,thisbookfocusesonthecaseof γ> 0,i.e.,theFréchetdistribution in (1.5).
Forthestudyofextremecomovementoffinancialmarkets,multivariate extremevaluetheoryisneeded,whichisbasedontheassumptionthatthere existconstantvectors an > 0, bn ∈ Rd andanon-degenerate d -dimensional distributionfunction H suchthat
where {Zi = (Zi1 , ··· , Zid )T , i ≥ 1} isasequenceofi.i.d.randomvectors in Rd withacommondistributionfunction F (z1 , ··· , zd ) andmarginal distributions Fj (zj ) for j = 1, ··· , d .Throughoutweuse AT todenotethe transposeofthevectorormatrix A
Liketheunivariateextremevaluetheory, H in (1.7) iscalledamultivariateextremevaluedistributionand F issaidtobeinthedomainofattraction of H (notation: F ∈ D(H ) ).Sincetheconvergenceofthejointdistributions formultivariateextremesimpliestheconvergenceofthemarginaldistributions,themarginaldistribution Hi of H mustbeoneofthethreetypes ofextremevaluedistributions.Also H isacontinuousfunctionsinceits marginaldistributionsarecontinuous.
Alsonotethat (1.7) isequivalentto lim n→∞ F n (an1 z1 + bn1 , , and zd + bnd ) = H (z1 , , zd ) (1.8)
forall (z1 , ··· , zd ) ∈ Rd ,whichisequivalentto
lim n→∞ n{1 F (an1 z1 + bn1 , ··· , and zd + bnd )}=− log H (z1 , ··· , zd )
Fj ∈ D(Hγj ) with Hγj givenin (1.6) and γj ∈ R (1.9) for j = 1, ··· , d ,andthefollowingdependencecondition
limn→∞ n{1 F (U1 (nx1 ), , Ud (nxd ))}=− log H ( x γ1 1 1 γ1 , ,
d d 1 γd ) =: l (x1 , ··· , xd ), (1.10)
where Ui (x) = Fi (1 1/x) for i = 1, ··· , d .Thelimitingfunction l (x1 , , xn ) in (1.10) iscalleda taildependencefunction,whichis ahomogeneousfunctionsatisfying
l (tx1 , ··· , txd ) = t 1 l (x1 , ··· , xd ) forany t > 0.
• Cauchydistribution: F (x) = ∞ x 1 π(1+t 2 ) dt for −∞ < x < ∞ .Thetailindex is1.
• t-distribution: F (x) = ∞ x Γ((ν +1)/2) Γ(ν/2)√νπ (1 + t 2 ν ) ν +1 2 dt for −∞ < x < ∞ ,where ν> 0.Thetailindexis ν
• Burrdistribution: F (x) = (1 + xb ) a for x ≥ 0,where a, b > 0.Thetail indexis ab (notation:Burr(a, b)).
• Log-Gammadistribution: F (x) = ∞ x αβ Γ(β) (log t )β 1 t α 1 dt for x ≥ 1, where α> 0,β> 0.Thetailindexis α .
• Assumethat ξ and η aretwoindependentrandomvariables, η isaheavy tailedrandomvariable,and ξ ≥ 0.Then ξη canbeaheavytailedrandom variableaccordingtothefollowingBreiman’slemma.
Breiman’sLemma: therandomvariable ξη hasaheavytailwithindex α andsatisfies lim t →∞ P (ξη> t ) P (η> t ) = E (ξ α ) if lim t →∞
t →0 F (tx) F (t ) = x 1/α forall x > 0, (2.2) whichisequivalentto
where F (x) = F (1 x) for0 < x < 1,and F denotesthegeneralized inversefunctionof F asdefinedin (1.1).Hencecondition (2.1) (i.e., F (x) ∈ RV ∞ α )isequivalentto F (x) ∈ RV 0 1/α ,andisequivalentto x1/α F (x) ∈ RV 0 0
Inordertospecifyanapproximationratein (2.2) or (2.3),whichplays animportantroleinderivingtheasymptoticpropertiesofestimatorsfor thetailindex α andsomerelatedquantitiessuchasahighquantileand anextremetailprobability,onecouldassumethatthereexistafunction c (x) ≡ 0andafunction A(t ) → 0withaconstantsignnearzerosuchthat
t →0 (tx)1/α F (tx) t 1/α F (t ) A(t ) = c (x) forall x > 0.
Inthiscase, t 1/α F (t ) iscalleda Π -variation,andbyTheoremB.2.1of deHaanandFerreira [27],wecouldassumethatthereexistsome ρ ≥ 0 andafunction A(t ) ∈ RV 0 ρ with limt →0 A(t ) = 0suchthat
Herecondition (2.5) isalsocalleda secondorderregularvariation conditionforthefunction t 1/α F (t ) .
Example2.1. Suppose1 F (x) = Cx α 1 + Dx ρα + o(x ρα ) forsome α> 0, C > 0, D = 0,ρ> 0as x →∞ .Then
i.e., (2.4) holdswith A(
.
Although (2.2), (2.4) and (2.5) aredefinedforeachfixed x > 0,they dohaveasortofuniformconvergencepropertyasdemonstratedbythe followinginequalities.Thistypeofuniformconvergenceplaysausefulrole inderivingtheasymptoticbehaviorofestimatorsandtestsinanalyzing extremes.
Proof. Bytherepresentationtheoremofaregularvariation,thereexist a t1 > 0,functions b(t ) and c (t ) with lim t →0 c (t ) = c0 ∈ (0, ∞) and lim t →0 b(t ) =−ρ
suchthatforall0 < t < t1
f (t ) = c (t ) exp
1 t b(s) s ds .
Hence f (tx) f (t ) = c (tx) c (t ) exp t tx b(s) s ds , whichcanbeusedtoprove (2.6) and (2.7) straightforwardly.
Proof. When ρ> 0,wehave f (0) := limt →0 f (t ) isfinite, a(t ) := f (t ) f (0) ∈ RV 0 ρ and a(t )/A(t ) → ρ 1 as t → 0,whichimply (2.9) byusing (2.7) andwritingthat
When ρ = 0,fora t1 > 0,define a(t ) = f (t ) t 1 t1 t f (s) ds.Thenwe have a(t ) A(t ) → 1, a(t ) ∈ RV 0 0 , and f (t ) = a(t ) + t1 t a(s) s ds. (2.10)
f (tx) f (t ) A(t ) log x = a(tx)/a(t ) 1 A(t )/a(t ) + 1 x
When ρ = 0and γ< 0,itfollowsfromdeHaanandStadtmüller [28] that
as t → 0.Applying (2.9) with f (x) = H (x) c
log(x) , (2.11) follows from (2.14) andthefollowingexpression
1 t h(s) s ds anditfollowsfromdeHaanandStadtmüller [28] that
Write H (tx) H (t ) A(t ) log x B(t ) 1 2 (log x)2
h(tx) h(t ) A(t )B(t ) log x 1 x 1 s h(ts) h(t ) A(t )B(t ) log s ds h(t ) A(t ) A(t )B(t ) 1 log x (2.17)
Thenitfollowsfrom (2.5), (2.16) and (2.17) that h(t ) A(t ) A(t )B(t ) → 1as t → 0. (2.18)
Applying (2.9) with f (x) = h(x) , (2.11) followsfrom (2.17) and (2.18).
Sincesomestatisticsinanalyzingextremesareconstructedintermsof logarithmsofdata,itbecomesconvenienttoexpress (2.2) and (2.4) as lim t →0 log F (tx) log F (t ) =− 1 α log x forall x > 0, (2.19)
and
(
x > 0, (2.20)
respectively.However,when ρ = γ , (2.5) maynotimplyasecondorderregularvariationconditionfor log F (t ) ;seeTheoremAofDraisma etal. [30].ThefollowingtheoremisslightlydifferentfromTheoremAof Draismaetal. [30],andshowsthat (2.5) doesimplyacorrespondingresult for log F (t ) underanadditionalcondition.Thistheoremisusefulinthe studyofbiascorrectedtailindexestimation.
Theorem2.1. Suppose (2.5) holdswith ρ> 0 and γ ≥ 0.Furtherassume
Let U1 , ··· , Un beindependentandidenticallydistributedrandomvariableswithuniformdistributionon (0, 1) ,andlet Un,1 ≤···≤ Un,n denote theorderstatisticsof U1 , ··· , Un .Therefore,
(u) = 1 n n i=1 I (Ui ≤ u) and αn (u) = √n {Gn (u) u}
arecalledthe empiricaldistributionfunction and empiricalprocess, respectively.BytheChibisov–O’Reillytheorem, sup 0<u<1 |αn (u)| u1/4 = Op (1) and sup
where αn (u ) denotestheleft-handlimitof αn (u) .Italsofollowsfrom ShorackandWellner [98],Page404that sup Un,1 ≤u≤1 u Gn (u) = Op (1), sup 1 Un,n ≤u≤1 u 1 Gn (1 u) = Op (1), (2.24) sup 0<u<1
for k = 1, ··· , n,and Qn (0) = Un,1 , whichiscalledthe quantilefunction.Further βn (s) = √n {Qn (s) s} is calledthe quantileprocess.AgainitfollowsfromCsörg ˝ oetal. [25] that forany ν ∈[0, 1/2) and λ> 0
where Bn (s) isgivenin (2.26).
Theorem2.2. Definethe tailempiricalprocess as
,k (u) = √k
whereksatisfies k = k(n) →∞
Thenforany ν ∈[0, 1/2)
whereWn (u) = n k Bn ( k n u) withBn givenin (2.26).
Proof. Notethat
,
Hence (2.29) followsfrom (2.26) and (2.28) when ν iscloseto1/2and u ≥ nUn,1 /k.When u < nUn,1 /k,wehave
≤ k 1/2+δ (nUn,1 )1 δ + u 1/2 δ | n ku Bn ( k n u)| = op (1).
where k satisfies (2.28).Thisistheso-calledHillestimatorintheliterature (seeHill [56]).AnotherusefulwaytoderivetheaboveHillestimator ˆ α(k) isviamaximizingacensoredlikelihoodfunctionasfollows.
Define δi = I (Xi > T ) forahighthreshold T andtemporarilyassume theconditionaldistributionof Xi given δi = 1is1 cx α .Here I (A) denotes theindicatorfunctionoftheset A.Thenacensoredlikelihoodfunctionfor (Xi ,δi )T n i=1 canbewrittenas
=1
whichismaximizedat
ThereforetheHillestimator ˆ α(k) in (2.31) isobtainedbytaking T = Xn,n k inthesecondequationof (2.33).
