Maximum lq-likelihood estimator of the heavy-tailed distribution parameter

Kouide , Mohammed Tidha; Idiou, Nes ine; Toumi, Samia; Bena ia, Fa ah
A icle
Maximum lq-likelihood es ima o o he hea y- ailed
dis ibu ion pa ame e
C oa ian Re iew o Economic, Business and Social S a is ics (CREBSS)
P o ided in Coope a ion wi h:
C oa ian S a is ical Associa ion (CSA), Zag eb
Sugges ed Ci a ion: Kouide , Mohammed Tidha; Idiou, Nes ine; Toumi, Samia; Bena ia, Fa ah (2024) :
Maximum lq-likelihood es ima o o he hea y- ailed dis ibu ion pa ame e , C oa ian Re iew o
Economic, Business and Social S a is ics (CREBSS), ISSN 2459-5616, C oa ian S a is ical Associa ion
(CSA), Zag eb, Vol. 10, Iss. 2, pp. 29-48,
h ps://doi.o g/10.62366/c ebss.2024.2.003
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C oa ian Re iew o Economic, Business and Social S a is ics 29
CREBSS 10(2):29–48
Maximum lq–likelihood es ima o o he hea y– ailed dis ibu ion
pa ame e
Mohammed Ridha Kouide 1,*, Nes ine Idiou 2, Samia Toumi 3and
Fa ah Bena ia4
1,3,4 Mohamed Khide Uni e si y o Bisk a, Facul y o Exac Science
and Na u al and Li e Science, Alge ia Q
2Salah Boubnide Uni e si y o Cons an ine 3, Facul y o P ocess
Enginee ing, Alge ia
ARTICLE TYPE
P elimina y communica ion
ARTICLE INFO
Recei ed: Sep embe 7, 2024
Accep ed: No embe 5, 2024
DOI: 10.62366/c ebss.2024.2.003
JEL: C13, C46
SUMMARY
S udying he ex eme alue heo y (EVT) in ol es mul iple main ob-
jec i es, among hem he es ima ion o he ail index pa ame e . Some
es ima ion me hods a e used o es ima e he ail index pa ame e like
maximum likelihood es ima ion (MLE). Addi ionally, he Hill es ima-
o is one ype o maximum likelihood es ima o , which is a mo e
obus wi h a la ge sample han a small sample. This esea ch p o-
poses he cons uc ion o an al e na i e es ima o o he pa ame e o
he hea y– ailed dis ibu ion using he maximum lq–likelihood es i-
ma ion (MLqE) app oach in o de o adap he ML and Hill es ima-
o wi h he small sample. Fu he mo e, he maximum lq–likelihood
es ima o asymp o ic no mali y is es ablished. Mo eo e , se e al sim-
ula ion s udies in o de o compa e he MLq es ima o wi h he ML
es ima o s a e p o ided. In he excesses o e high sui able h eshold
alues he numbe o he la ges obse a ion kwill lead o an e icien
es ima e o he Hill es ima o . Fo his, selec ion o kin he Hill es ima-
o was in es iga ed using he me hod o he quan ile ype 8 which is
e ec i e wi h he hyd ology da a. The pe o mance o he Hill es ima-
o and he lq–Hill es ima o is subsequen ly compa ed by employing
eal elies wi h he dis ibu ion o hyd ology da a.
KEYWORDS
excesses o e h eshold, ex eme alue index, hea y– ailed dis ibu ion, max-
imum lq–likelihood es ima o
1. In oduc ion
Conside ing X1,X2, . . . , Xno independen ly and iden ically dis ibu ed (iid) andom a i-
ables ( ) de ined o e some p obabili y space (Ω;A;P), wi h cumula i e dis ibu ion unc-
ion (cd ) F. We a e in e es ed in he p obabili y ha he maximum is no beyond a ce ain
h eshold x. This p obabili y is gi en by
P(max (X1,X2, . . . , Xn)≤x)=Fn(x). (1)
∗Co esponding au ho
©2024 Copy igh o his a icle is e ained by he au ho (s)
This is an open access a icle unde he CC BY–NC–ND 4.0 license
30 Kouide , Idiou, Toumi & Bena ia
As i is well known, when we a e in e es ed in he cen al pa o a sample, he cen al
limi heo em (CLT) gi ing he asymp o ic law o he sum o obse a ions which says ha
he sampling dis ibu ion o he mean will always be no mally dis ibu ed as n→+∞. On
he o he hand, i we wan o s udy he ex eme alues o his sample, he CLT p esen s
only li le o in e es . Ins ead, we use a esul es ablishing he asymp o ic dis ibu ion o
he maximum o he sample. This esul is s a ed unde EVT as demons a ed pa icula ly
by Gnedenko (1943). The EVT gi es he condi ions unde which he e exis sequences o
no malizing cons an s an>0 and bn>0 such ha
lim
n→+∞Fn(anx+bn)=Gγ(x). (2)
Gγ(x)is so–called he ex eme alue dis ibu ion, de ined by
Gγ(x)=(exp −(1+γx)−1/γ, i γ=0
exp (−exp (−x)) , i γ=0(3)
whe e Gγ(x)is a well–de ined non–degene a e cd . This law depends only on he pa ame e
γ∈Rcalled he ex eme alue index o he ail index, o he shape pa ame e . Acco ding o
he sign o γ, he e a e h ee domains o a ac ion ha a e de ined om Gγ(x)depending on
he ail index; among hem is domain a ac ion o F éche . Also, i is e e ed o as hea y–
ailed dis ibu ion. I con ains laws whose su i al unc ion dec eases as a powe unc ion
like dis ibu ions o Pa e o, S uden , Cauchy, e c. So he hea y ailed limi dis ibu ion is he
F éche dis ibu ion (Balkema and de Haan,1974), which is de ined by
Gγ>0(x)=e−x−1/γ. (4)
As i is known and associa ed wi h EVT, he cha ac e iza ion o he domains o a ac ion
makes ex ensi e use o he no ion o unc ions wi h egula a ia ions which we de ine below.
Le X1,X2, . . . , Xnbe a iid sequence o a non–nega i e Xo e some p obabili y space
(Ω;A;P), wi h cd F. We assume ha he dis ibu ion ail F=1−Fis egula ly a ying a
in ini y, wi h index (−1/γ), no a ion: F∈RV(−1/γ). Tha is
lim
→+∞
F( x)
F( )=x−1/γ, o any x>0. (5)
A dis ibu ion unc ion Fbelongs o he domain a ac ion o F éche i and only i
F∈RV(−1/γ). Hence, he ail beha es app oxima ely as a powe unc ion x−1/γ. This im-
plies ha he dis ibu ion o he maximum has a one– o–one ela ionship wi h he shape
pa ame e . Then, we will ake hea y ailed o mean sub–exponen ial (de ini ion below), bu
se e al de ini ions exis in he li e a u e. Fo he con enience o he eade , we also ske ch
o he common de ini ions and, whe e possible, ela e hem o one ano he . Sub-exponen ial
dis ibu ions exhibi one o he gene al p ope ies expec ed o hea y– ailed dis ibu ions on
he le el o agg ega e losses, namely ha he ail o he maximum de e mines he ail o he
sum. All o he dis ibu ions conside ed he e a e sub–exponen ial.
Le Fbe a cd wi h suppo in [0; +∞[. Then Fis sub–exponen ial i , o all n≥2,
lim
x→+∞
Fn(x)
F(x)=n, o any x>0. (6)
Maximum lq–likelihood es ima o o he hea y– ailed dis ibu ion pa ame e 31
Then Fn(x)≈nF (x)as x→+∞. Subexponen iali y implies ano he p ope y ha is
some imes aken as he de ini ion o hea y ail, i.e. he ail decays mo e slowly han any
exponen ial unc ion. Wi h he no a ion as abo e, he p ecise o mula ion is ha o all >0,
lim
→+∞e x F(x):=∞. (7)
An impo an subclass o sub–exponen ial dis ibu ions consis s o egula ly a ying unc-
ions. Fo a egula ly a ying wi h ail index γ>0 , all momen s o he associa ed highe
han γ>0 will be unbounded (Emb ech s e al.,1997).
The pa ame e o in e es is γ>0 is he ail index o F. Now o F∈RV(−1/γ)we ake
as F(x)=x−1/γ. Then we can check o Fis sub–exponen ial ha
lim
→+∞
lim
x→+∞
1
Fn( )
∞
Z
Fn(x)
xdx
=1
F( )
∞
Z
F(x)
xdx :=γ. (8)
Conside X1,n≤X2,n≤. . . ≤Xn,n he o de s a is ics o X1,X2, . . . , Xn. Le ’s eplace he
dis ibu ion Fby i s empi ical e sion Fnand by Xn−k,n. Thus, we ind he Hill es ima o
(Hill,1975) de ined by:
b
γH
Xn−k,n:=1
Fn(Xn−k,n)
∞
Z
Xn−k,n
Fn(x)dx
x. (9)
The Hill es ima o can only be used o dis ibu ions belonging o he F éche domain. The
Hill es ima o , which is a ype o ML es ima o , is he mos common es ima o s o he ail in-
dex o hea y– ailed dis ibu ions; is p obably he mos s udied es ima o in he li e a u e. As
ag eed ha , Hill es ima o and ML es ima o a e goods wi h he la ge sample is suscep ible
o be biased. Bu i we use e y small k, bo h es ima o s ha e a la ge a iance.
In his a icle, we in es iga e a new class o pa ame ic es ima o s based on he q–en opy
unc ion p oposed by Ha da and Cha á (1967). I has been o conside able in e es in di -
e en domains o applica ion like physics, inance and biomedical sciences. As well, Al un
and Smola (2006) ha e seen ha he classical maximum en opy is dual o MLE. Di o, Fe -
a i and Yang (2010) p oposed a new pa ame ic es ima ion me hod based on he q–en opy
unc ion, he MLqE whe e qis called he dis o ion pa ame e . Also, hey ha e p o en o be
a e y use ul me hod when es ima ing high–dimensional pa ame e s and small ail p obabil-
i ies. This is impo an in many applica ions whe e he numbe o a ailable obse a ions is
no g ea . They ha e shown ha MLqE becomes he MLE wi h q=1.
Since o la ge sample he ML and Hill es ima o s a e a leas as p ecise as any o he es-
ima o s. Howe e , o a mode a e o small sample size he MLq es ima o can o e d ama ic
imp o emen in mean squa ed e o a he expense o a sligh inc ease in bias.
This pape has been o ganized as ollows. In Sec ion (1) i is p esen ed as an in oduc-
ion he asymp o ic dis ibu ion o he maximum o he sample unde he EVT and especially
he hea y– ailed dis ibu ion o he F éche dis ibu ion. We also ga e a de ini ion abou
sub–exponen ial. As o la e , we will ocus on p esen ing he new es ima e abou he ail
index o hea y– ailed dis ibu ions, i.e. MLq es ima o . Nex , wi h Sec ion (2), we p esen he
basic asymp o ic no mali y o MLq es ima o wi h hei consis en o exponen ial amilies
which we in oduce in he same sec ion. In Sec ion (3), we p esen a simula ion s udy wi h
Pa e o dis ibu ion o compa e he MLqE wi h MLE. Also, he eal da a a e u ilized o illus-
a e he use ulness o he F éche dis ibu ion as he dis ibu ion o hyd ology da a. Finally,
concluding no es a e p o ided in Sec ion (4).
32 Kouide , Idiou, Toumi & Bena ia
1.1. Adap i e ML and Hill es ima o s o small sample
In he domain o hea y– ailed dis ibu ions, he s a is ic o EVT ansla es in o a semi-
pa ame ic es ima ion p oblem. Indeed, i Fbelongs o he F éche domain, hen Fis o
he o m x−1/γℓ(x)wi h ℓa slowly a ying unc ion i.e, lim
→+∞
ℓ( x)/ℓ(x):=1. This pape
concen a es on he dis ibu ions ha ha e a egula ly a ying ail,
F(x)
x−1/γℓ(x):=1, as x→+∞,γ>0 (10)
No e ha Gγ>0(x)sa is ies (10). He e (1/γ)is he index o egula a ia ion, o he ail
index. Then Fhas a pa ame ic pa x−1/γdepending only on γ, and a non–pa ame ic pa
ℓ. Fo a eal >0 i ’s clea ha we deduce
F(x):=F( )x
−1/γ(11)
Le X1,X2, . . . , Xnbe a sequence o iid om dis ibu ion unc ion (d ) Fand le X1,n≤
X2,n≤. . . ≤Xn,ndeno e he o de s a is ics co espondence. We deno ing he numbe o
absolu e excesses o e by k o a he he la ges obse a ions (Xn−k,n, . . . , Xn,n)whe e in he
asymp o ic se ing k=knan in e media e sequence, ha is, kn→∞and kn/n→0 as n→∞.
Then, see Haan and Fe ei a (2006) Lemma 3.4.1, he join dis ibu ion o (Xn−k,n, . . . , Xn,n)
o k=1, . . . , n−1 be he d gi en by
F (x)=P(X≤x|X> )=F(x)−F( )
1−F( ) o x> (12)
Such ha F (x)=1−F (x)we can ew i e (12) o x> as
F (x)=F(x)
F( )(13)
Since F(x)is gi en by (11), also we can ew i e F (x)gi en in (13) o x> by
F (x)=x
−1/γ(14)
I ’s easy o checked ha unde (5) ha F ∈RV(−1/γ).
The pa ame e o he F (x)can be es ima ed using s anda d me hods such as he MLE.
Ano he es ima ion me hod is he MLqE which based on q–o de en opy. The q–o de en-
opy which is p o ided by Ha da and Cha á (1967), has he unc ion
Lq(u)=(u1−q−1
1−q o q<1
log u o q=1(15)
whe e uis p obabili y densi y unc ion(pd ) and qcalled he dis o ion pa ame e . The MLqE,
which is p oposed by Fe a i and Yang (2010), use he Lq(u) unc ion ins ead o he log–
likelihood unc ion as in he MLE. Obse ed ha when q=1, we ha e he MLq es ima o
app oaches as he ML es ima o app oaches. The MLqE me hod educes he e ec o ex eme
obse a ions on pa ame e es ima es using q. The choice o qis ano he di icul p oblem in

Maximum lq–likelihood es ima o o he hea y– ailed dis ibu ion pa ame e 33
MLqE es ima ion. In his esea ch, we ake q=1−1
kas gi en by Fe a i and Yang (2010) and
we no e ha q→1 as k→∞. Then, he MLqE o γ>0 is gi en by
b
γ=a g max
k
∑
i=1
Lq( (x)) (16)
wi h
Lq( (x)) =( (x)1−q−1
1−q o 0 <q<1
log u o q=1(17)
whe e (x)is he pd o F (x)gi en in (14). Thus de ense o γ>0 by
(x)=1
γ
1
x
−1/γ−1(18)
As is known, MLq es ima o o γ>0 can be ob ained by maximizing ∑k
i=1Lq( (x))
wi h espec o he pa ame e γ>0. Then, o 0 <q≤1 we ge
∂
∂γ Lq( (x)) :=∂
∂γ log ( (x)) (x)1−q(19)
Then, he equa ions om (19) a e hen gi en in e m o he pa ial de i a i e espec o
γby:
∂
∂γ
k
∑
i=1
Lq( (Xi)) :=
k
∑
i=1
1
γ2(log (Xi)−log ( )−γ) (Xi)1−q=0 (20)
Nex , we can de ine he es ima o o γ>0 by
b
γMLq
=
k
∑
i=1
wi(log (Xi)−log ( ))
k
∑
i=1
wi
wi h wi= (Xi)1−q(21)
The e mo e, i we ake =Xn−k,nwe ind a new Hill es ima o called lq–Hill es ima o
which is de ined by
b
γHLq
Xn−k,n=
k
∑
i=1
wn−i+1,nlog Xn−i+1,n−log (Xn−k,n)
k
∑
i=1
wn−i+1,n
wi h wn−i+1,n= Xn−i+1,n1−q(22)
I is clea ha i q=1 gi es us he classic he MLE. We ge ML es ima o o γ>0 which
is gi en by:
b
γML
=1
k
k
∑
i=1
log (Xi)−log ( )(23)
And by hei equi alen s in he o de s a is ic, we ind he o mula o he Hill es ima o
wi h =Xn−k,nby
b
γH
Xn−k,n=1
k
k
∑
i=1
log Xn−i+1,n−log (Xn−k,n)(24)
34 Kouide , Idiou, Toumi & Bena ia
Also, he Hill es ima o was ound o be e y sensi i e o he choice o index k. Because
choosing he op imal alue o he kindex will lead o an e ec i e es ima e o Hill es ima o ,
he choice o kis ano he di icul p oblem and he e a e many esea ches on his. Howe e , in
his a icle, we will p esen in sub–sec ion (3.2) a me hod ha allows calcula ing he numbe
kbased on he me hod o he quan ile ype 8 wi h he hyd ology da a. I is impo an o
men ion ha Hill es ima o is a consis en o he ail index and asymp o ically no mal wi h
mean γand a iance γ2/k. Hence, i one uses a e y small k, he es ima o has la ge a iance,
howe e , o e y la ge k, he es ima o is likely o be biased i.e asymp o ically no mal wi h
mean 0 and a iance γ2. This is why i is good wi h la ge sample also, ML es ima o is
e ec ed wi h la ge sample. In his esea ch, we ha e w i en his ow es ima o wi h he
MLqE. We can show ha o q=1−1
kwe ew i e he es ima o s b
γMLq
and b
γHLq
Xn−k,nas
b
γMLq
=
k
∑
i=1
(Xi)1/k(log (Xi)−log ( ))
k
∑
i=1
(Xi)1/k
(25)
and
b
γHLq
Xn−k,n=
k
∑
i=1
Xn−i+1,n1/klog Xn−i+1,n−log (Xn−k,n)
k
∑
i=1
Xn−i+1,n1/k
(26)
Recall ha when qis chosen co ec ly o small samples, he MLqE can ade bias o
accu acy success ully. This leads o a signi ican dec ease in he mean squa ed e o . The e a e
mo e o la ge sample i (k→∞)q=1−1
ko q→1 we ocus on a necessa y and su icien
condi ion, o ensu e a p ope asymp o ic no mali y and e iciency o MLqE is es ablished.
This is wha will be discussed in he nex sec ion.
2. Main esul
In his sec ion, we discuss he basic asymp o ic p ope ies o he MLq es ima o when he
deg ee o dis o ion depends on he amoun o in o ma ion a ailable in he sample. Such
p ope ies will be used la e on o de i e ou main esul s. In he eminde o he pape , ou
analysis ocuses on he dis ibu ions belonging o he exponen ial amily. In pa icula , we
conside pd o F (x)gi en on (22) in he o m
(x)=exp 1
γb(x)−c(x)−A(γ)(27)
Fo
b(x)=log x
and c(x)=log (x),A(γ)=log (γ)
In addi ion, we de ine ψγ(x)=1
γb(x)−c(x)−A(γ). So ha (x)and log (x)wi h
i s de i a i e unc ion espec o γexp essed espec i ely as
(x)=exp ψγ(x)and ∂
∂γ (x)=−1
γ2b(x)−1
γexp ψγ(x), (29)
Maximum lq–likelihood es ima o o he hea y– ailed dis ibu ion pa ame e 35
and
log (x)=ψγ(x)and ∂
∂γ log (x)=−1
γ2b(x)−1
γ. (30)
Un il, o γ>0 we can check ha
A(γ)=log
+∞
Z
exp 1
γb(x)−c(x)dx =log γ, (31)
is he cumula i e gene a ing unc ion (o log no malize) and di e en ia ing k imes gi es
∂k
∂kγA(γ)=(−1)k−1(k−1)!
γk, (32)
Th oughou he cou se o he discussion he ue pa ame e will be deno ed by γ>0
. Nex , we explo e consis ency, which is a basic equi emen o a good es ima o . Le , o
0<q≤1
φk(γ)=1
k
k
∑
i=1
∂
∂γ Lq( (Xi)) :=1
k
k
∑
i=1−1
γ2b(x)−1
γe(1−q)1
γb(x)−c(x)−A(γ)(33)
The MLqE is ound by se ing φk(γ)=0 and sol ing o γ. Since o q=1 and γ=0 in
he abo e exp ession gi es he usual MLE equa ion
1
k
k
∑
i=1−1
b
γMLq b(x)−1=0 (34)
Theo em 2.1. Le X1,X2, . . . , Xnbe a sequence o iid om d F and le X1,n≤X2,n≤. . . ≤Xn,n
deno e he o de s a is ics co espondence. Conside ing he la ges obse a ions (Xn−k,n, . . . , Xn,n)
o k =1, . . . , n−1 om he d F (x)and pd (x)as (27). Then, o any MLq es ima o o
b
γ=a g max ∑k
i=1Lq( (x)) as k →∞we ha e ha
b
γP
→γ.
wi h Lq( (x)) is gi en in (15) and q →1as k →∞.
P oo . De ine, φ(γ)=Eγh∂
∂γ log (x)i. Then we can ew i e φ(γ)=Eγh−1
γ2b(x)−1
γi.
Now, we wan o show ha o all φk−φ→0 as k→∞whe e
φk−φ=1
k
k
∑
i=1−1
γ2b(x)−1
γe(1−q)ψγ(x)−Eγ−1
γ2b(x)−1
γ(35)
Then we ha e
|φk−φ|=
1
k
k
∑
i=1−1
γ2b(x)−1
γe(1−q)ψγ(x)−1+1
k
k
∑
i=1−1
γ2b(x)−1
γ−Eγ−1
γ2b(x)−1
γ
hus
≤
1
k
k
∑
i=1−1
γ2b(x)−1
γe(1−q)ψγ(x)−1+
1
k
k
∑
i=1−1
γ2b(x)−1
γ−Eγ−1
γ2b(x)−1
γ
(36)
36 Kouide , Idiou, Toumi & Bena ia
By he law o la ge numbe s we ind

1
k
k
∑
i=1−1
γ2b(x)−1
γ−Eγ−1
γ2b(x)−1
γ→0 (37)
Then he inequali y (35) becomes
|φk−φ|=
1
k
k
∑
i=1−1
γ2b(x)−1
γe(1−q)ψγ(x)−1(38)
By he Hölde ’s inequali y, we can ew i e (38) as ollows
|φk−φ|≤
u
u
1
k
k
∑
i=1e(1−q)ψγ(x)−12
u
u
1
k
k
∑
i=1−1
γ2b(x)−1
γ2
(39)
And unde Jensen’s inequali y we ha e
|φk−φ|≤1
4
1
k
k
∑
i=1e(1−q)ψγ(x)−121
k
k
∑
i=1−1
γ2b(x)−1
γ2
(40)
Fo he le side om he p e ious inequali y and by he basic ac ha (1+u)2≤e2u o any
eal numbe uwe ew i e
1
γ2k
k
∑
i=1−1
γb(x)−12
≤1
γ2k
k
∑
i=1
e2
γb(x)=1
γ2k
k
∑
i=1x
−2
γ(41)
Then we ha e
1
γ2k
k
∑
i=1
e2
γb(x)=1
γ2k
k
∑
i=1
F (x)2<∞(42)
And o he igh side o he inequali y ha unde he law o la ge numbe s we ge
1
k
k
∑
i=1e(1−q)ψγ(x)−12→Eγe(1−q)ψγ(x)−12(43)
Wi h q→1 as k→∞we ha e
Eγe(1−q)ψγ(x)−12→0
Finally we ob ained ha φk→φas k→∞.
As de ined in heo em (2.1), he MLqE is a consis en es ima o o γ. Al hough o a ixed
q=1. The MLqE is clea ly asymp o ically biased; a clea imp o emen is ob ained by le ing
he dis o ion pa ame e depends on he sample size. To ob ain he asymp o ic no mali y o
MLqE, we shall discuss he educ ion in e ms o a iance achie ed by conside ing a sligh ly
di e en a ge pa ame e o each γ>0 and k≥1.
On pa icula , we conside b
γMLq he alue such ha
E∂Lq( (x))
∂b
γMLq =0 (44)
Maximum lq–likelihood es ima o o he hea y– ailed dis ibu ion pa ame e 43
4. Concluding no es
Hea y– ailed dis ibu ions would be a good al e na i e o he dis ibu ions ha a e used in
economics, eliabili y, su i al analysis and so on. The pa ame e s o his dis ibu ion ha e
been es ima ed using MLE me hod. Recen ly, ML and Hill es ima o s a e he mos es ima o s
used o es ima e he pa ame e o he ail beha io . Fo oo la ge sample, ML and Hill
es ima o s a e likely o be good and obus me hods should be used o es ima e he shape
pa ame e . Howe e , ML and Hill es ima o s canno be compa ible wi h he small sample.
In his pape , we ha e used he MLq es ima ion me hod o es ima e he shape pa ame e o
he hea y– ailed dis ibu ion o small sample. We ha e ca ied ou o adap he ML and Hill
es ima o s in he case o small sample, b
γMLq
and b
γHLq
Xn−k,nas in (25) and (26), espec i ely. Since
b
γMLq
and b
γHLq
Xn−k,na e bo h esul om he MLqE, we es ablish he asymp o ic no mali y o he
MLq es ima o o Hea y ailed dis ibu ions pa ame e γ>0. Acco ding o co olla y (2.1) o
e y la ge k, he MLq es ima o is likely o be biased. And i q=1, he MLqE becomes MLE
and he no mali y asymp o ic o MLq es ima o becomes as ML es ima o . Since he MLq
es ima o depends on he o de s a is ical index k o hea y ailed dis ibu ions, we used an
app oach o selec ing kwhich is de ined in (62) by using ype 8 quan ile es ima o Q8(0.7)
in (60) om he s able egion o Hill plo , which will be a mo e lexible al e na i e o use in
Hill’s es ima o . This app oach will be an app oxima ion o he ex eme alue index a he
ail end o he dis ibu ion o hyd ology da a.
Appendix
Fi s , we calcula e E∂Lq( (x))
∂e
γ2. Le ’s conside
E"∂Lq( (x))
∂e
γ2#=1
γe
γ6 1
γ+2
e
γe
γ2q −2q
e
γ
∞
Z
(log (Xi)−log ( )−e
γ)2e−[2eα(1−q)+α]log xdx (66)
wi h eα=1+1
e
γ. Pu ing log (x)=u, hen he p e ious equali y (66) becomes
E"∂Lq( (x))
∂e
γ2#=β
∞
Z
log u2−2u(log +e
γ)+(log +e
γ)2e−[2eα(1−q)+α−1]udu (67)
whe e β=1
γe
γ6 1
γ+2
e
γe
γ2q −2q
e
γis cons an wi h α=1+1
γ. Then, o e
γ=q/(α−q)we ha e
qeα=αwi h e
γ=1/ (eα−1). Fo e
θ=eα(q−1)+α−1=eα−1 hen 2eα(1−q)+α−1=
eα(2−q)−1.
Conside his decomposi ion
D( )=
∞
Z
log u2−2u(log +e
γ)+(log +e
γ)2e−[eα(2−q)−1]udu =I1−I2+I3(68)
whe e
I1=
∞
Z
log
u2e−[eα(2−q)−1]udu,

44 Kouide , Idiou, Toumi & Bena ia
I2=
∞
Z
log
2u(log +e
γ)e−[eα(2−q)−1]udu
And
I3=
∞
Z
log
(log +e
γ)2e−[eα(2−q)−1]udu
Beginning wi h I1=R∞
log u2e−[eα(2−q)−1]udu, unde in eg al by pa s we ha e
I1=
∞
Z
log
u2e−[eα(2−q)−1]udu =−1
eα(2−q)−1

hu2e−[eα(2−q)−1]ui∞
log −2
∞
Z
log
ue−[eα(2−q)−1]udu


(69)
Also wi h in eg al by pa s we ha e
∞
Z
log
ue−[eα(2−q)−1]udu =−1
eα(2−q)−1
"ue−[eα(2−q)−1]u+e−[eα(2−q)−1]u
eα(2−q)−1#∞
log 
(70)
Subs i u ing (26) in o I1and since eα(2−q)−1>0 we ge
I1=e−[eα(2−q)−1]log
eα(2−q)−1(log )2+2
eα(2−q)−1log +1
eα(2−q)−1 (71)
Then we go o I2=R∞
log 2u(log +e
γ)e−[eα(2−q)−1]udu, . By (26) and wi h eα(2−q)−1>0
we ha e
I2=2(log +e
γ)e−[eα(2−q)−1]log
eα(2−q)−1log +1
eα(2−q)−1(72)
And o I3=R∞
log (log +e
γ)2e−[eα(2−q)−1]udu, unde -pe o ming simple a i hme ic ope -
a ions, we ge
I3=(log +e
γ)2"−e−[eα(2−q)−1]u
eα(2−q)−1#∞
log
=(log +e
γ)2e−[eα(2−q)−1]log
eα(2−q)−1(73)
Unde (71)-(73) and D( )=I1−I2+I3we ha e
(eα(2−q)−1)e[eα(2−q)−1]log D( )=1+(eαe
γ(1−q))2
(eα(2−q)−1)2(74)
Then i gi es us ha
E"∂Lq( (x))
∂e
γ2#=βe−[eα(2−q)−1]log
eα(2−q)−1 1+(eαe
γ(1−q))2
(eα(2−q)−1)2!(75)
Since
β=1
γe
γ6 1
γ+2
e
γe
γ2q −2q
e
γ
Maximum lq–likelihood es ima o o he hea y– ailed dis ibu ion pa ame e 45
Nex , o eαe
γ=1+e
γwe can ind ha
E"∂Lq( (x))
∂e
γ2#=1
γ 1
γe2(q−3)log e
γ+2
e
γ(1−q)−(eα(2−q)−1)log
eα(2−q)−1 1+((1+e
γ) (1−q))2
(eα(2−q)−1)2!
Then
E"∂Lq( (x))
∂e
γ2#=1
γ 1
γe2(q−3)log e
γ−(q(eα−2)+1)log 1+((1+e
γ) (1−q))2
(eα(2−q)−1)3!(76)
Nex , we ha e o calcula e E∂2Lq( (x))
∂2e
γ. Then, conside
E" ∂2Lq( (x))
∂2e
γ!#=∂
∂e
γ
∞
Z
∂Lq( (x))
∂e
γ (x)dx (77)
Re e ing o equali y (48), we conclude ha
E" ∂2Lq( (x))
∂2e
γ!#=∂
∂e
γ1
γ 1
γe
γq−3 1
e
γ(1−q)1
e
θ2−1
e
θe−e
θlog 
=∂
∂e
γ1
γ 1
γ1
e
θ2−1
e
θe−e
θ+1
e
γ(1−q)log +(q−3)log e
γ
whe e e
θ=eα(1−q)+α−1 wi h eα=1+1/e
γimplies ∂e
θ/∂e
γ=−(1−q)/e
γ2and ∂eα/∂e
γ=
−1/e
γ2. Since
∂e−e
θ+1
e
γ(1−q)log +(q−3)log e
γ
∂e
γ=(q−3)
e
γe−e
θ+1
e
γ(1−q)log +(q−3)log e
γ
The e o e, we ha e
E" ∂2Lq( (x))
∂2e
γ!#=1
γ 1
γe−e
θ+1
e
γ(1−q)log +(q−3)log e
γ1−q
e
γ2e
θ2−2(1−q)
e
γ2e
θ3+q−2
e
θ−q−3
e
γe
θ2
Unde e
θ=eα(1−q)+α−1=1/e
γ. Then, we ge
E" ∂2Lq( (x))
∂2e
γ!#=1
γ 1
γe−1
e
γlog +(q−3)log e
γ((1−q) (1−2e
γ)+e
γ)(78)
Now, we ha e o coun he a iance o √kb
γMLq −e
γ/σ→N(0; 1)as k→∞. Wi h
σ2=
E∂Lq( (x))
∂e
γ2
Eh∂2Lq( (x))
∂2e
γi2(79)
And unde (76)-(78) we ge
σ2=1
γ −1
γe21
e
γqlog −(q(eα−2)+1)log 


1+((1+e
γ)(1−q))2
(eα(2−q)−1)3
((1−q) (1−2e
γ)+e
γ)2


(80)
46 Kouide , Idiou, Toumi & Bena ia
I ’s easy o ind ha
−(q(eα−2)+1)log +2
e
γqlog =1
γlog
Since qeα=αwe ind, γ=1/ (qeα−1). Then he a iance becomes
σ2=1
(qeα−1)
1+((e
γ+1) (1−q))2
(eα(2−q)−1)3((1−q) (1−2e
γ)+e
γ)2. (81)
wi h q=1 hen eα=αand e
γ=γ he a iance σ2w i e as
σ2=1
(α−1)
1
(α−1)3γ2(82)
Since (α−1)=1/γwe ha e σ2=γ2.
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48 Kouide , Idiou, Toumi & Bena ia
nis a
P ocjeni elj naj e´ce lq– je odos ojnos i pa ame a dis ibucije s eškim
epom
VRSTA ˇ
CLANKA
P e hodno p iop´cenje
INFORMACIJE O ˇ
CLANKU
P imljeno: 7. ujna 2024.
P ih a´ceno: 5. s udenog 2024.
DOI: 10.62366/c ebss.2024.2.003
JEL: C13, C46
SAŽETAK
P ouˇca anje eo ije eks emnih ijednos i (EVT) ukljuˇcuje
nekoliko gla nih cilje a, me ¯
du kojima je p ocjena pa ame a
epnog indeksa. Razliˇci e me ode p ocjene ko is e se za p oc-
jenu o og pa ame a, popu me ode naj e´ce je odos ojnos i
(MLE). Doda no, Hillo p ocjeni elj je jedan ip p ocjeni elja
naj e´ce je odos ojnos i koji je obusniji kod elikih uzo aka
nego kod malih. O o is aži anje p edlaže kons ukciju al-
e na i nog p ocjeni elja za pa ame a dis ibucije s " eškim
epom" ko is e´ci p is up naj e´ce lq– je odos ojnos i (MLqE),
kako bi se p ilagodili MLE i Hillo p ocjeni elj za male uzo ke.
Nadalje, u ¯
dena je asimp o ska no malnos p ocjeni elja na-
j e´ce lq– je odos ojnos i. Osim oga, p o edene su simulacijske
s udije kako bi se uspo edio MLq p ocjeni elj s MLE p ocjeni el-
jem. U sluˇcaje ima p eko aˇcenja isokih azina p ago a, p ik-
ladnih ijednos i, b oj naj e´cih opažanja k odi do e ikasne
p ocjene pomo´cu Hillo og p ocjeni elja. U u s hu, izbo k
kod Hillo og p ocjeni elja is ažen je me odom k an ila ipa 8,
koja se pokazala uˇcinko i om u analizi poda aka iz hid ologije.
Uˇcinko i os Hillo og p ocjeni elja i lq–Hillo og p ocjeni elja
za im je uspo e ¯
dena p imjenom s a nih poda aka s dis ibuci-
jom hid oloških ijednos i.
KLJU ˇ
CNE RIJE ˇ
CI
p eko aˇcenja iznad azine p aga, indeks eks emnih ijed-
nos i, dis ibucija s eškim epom, p ocjeni elj naj e´ce lq–
je odos ojnos i