Tes ing Copula Hypo hesis wi h Copula En opy
Jian MA∗
Hi achi China Resea ch Labo a o y
No embe 6, 2025
Abs ac
Tes ing copula hypo hesis is o undamen al impo ance in he applica-
ions o copula heo y. In his pape we p oposed a copula hypo hesis
es ing wi h copula en opy. Since copula en opy is a uni ied heo y in
p obabili y and he e o e es ing copula hypo hesis based on i can be
applied o any ypes o copula unc ion. The es s a is ic is de ined as
he di e ence o copula en opy o copula hypo hesis and ue copula en-
opy. We p opose he es ima ion me hod o he p oposed s a is ic and
wo special cases o Gaussian copula hypo hesis and A chimedean cop-
ula hypo hesis. We es he e ec i eness o he p oposed me hod wi h
simula ion expe imen s.
Keywo ds: Copula En opy; Copula; Hypo hesis Tes ; Gaussian Copula; A chimedean
Copula
1 In oduc ion
E alua ing he i ness o models o da a is a common p ac ice in scien i ic ac-
i i ies. Tes ing hypo hesis is one o he undamen al p oblems in s a is ics and
and has wide applica ions in e e y b anch o sciences.
Copula heo y is abou ep esen ing mul i a ia e dependence wi h copula
unc ions [1, 2]. As he co e esul o copula heo y, Skla ’s heo em [3] s a es
ha mul i a ia e densi y unc ion can be ep esen ed as a copula unc ion wi h
ma ginal unc ions as i s inpu s. The e a e many copula unc ion amilies a ail-
able o eal applica ions, such as Gaussian copula, Copula [4], A chimedean
copula, A chimax copula [5], Sibuya Copula [6], among o he s.
Modeling wi h copula unc ion is a widely used me hods in many scien-
i ic ields [7, 8, 9, 10, 11, 12] and hence es ing copula hypo hesis is impo -
an in hose p ac ices [13]. Many esea ch ha e been con ibu ed o copula
hypo hesis es ing, such as es ing Gaussian copula hypo hesis [14, 15], es -
ing A chimedeani y [16, 17], es ing symme y o copula [18]. These wo k
on es ing copula hypo hesis a e mainly ocusing on special ypes o copula
unc ion and a gene al me hod o any ypes o copula is needed. Kole, e
al [19] sugges using Kolmogo o -Smi no es and Ande son-Da ling es o
∗Email: ma[email p o ec ed]
1
selec copulas. G ønnebe g and Hjo [20] p oposed an AIC-like c i e ia o
copula model selec ion, named Copula In o ma ion C i e ia (CIC). Genes and
R´emilla d [21] p oposed o use C am´e - on Mises es and Kolmogo o -Smi mo
es o Goodness-o - i es ing o copula models.
Copula En opy (CE) is a ecen ly p oposed heo y in p obabili y. I de ined
he concep o CE as a special kind o Shannon en opy wi h copula unc ion
[22]. Copula unc ion ep esen s he dependence ela ionship be ween andom
a iables while CE measu es such ela ionship in a uni ied way. Con as o o he
dependence measu es based on copula, such as Spea man’s ρand Kendall’s τ,
CE has many good p ope ies, including non-nega i e, in a iance o mono onic
ans o ma ion, and equi alen o co ela ion ma ix unde Gaussiani y. CE has
been applied o hypo hesis es ing ecen ly, including mul i a ia e no mali y es
[23], wo-sample es [24], change poin de ec ion [25], and symme y es [26].
In his pape , we p oposed a copula hypo hesis es ing wi h copula en opy.
I can be used o any ypes o copula hypo hesis es ing. The es s a is ic is
de ined as he di e ence o CE o copula hypo hesis and ue CE. We gi e he
es ima ion me hod o he p oposed s a is ic and wo cases o Gaussian copula
hypo hesis and A chimdean copula hypo heses. We es he e ec i eness o he
p oposed me hod wi h simula ion expe imen s.
This pape is o ganized as ollows: Sec ion 2 in oduces he basic heo y
o CE, Sec ion 3 p esen s he p oposed es ing me hod, Sec ion 4 gi es he
es ima ion me hod o he p oposed s a is ic, simula ion expe imen s will be
p esen ed in Sec ion 5, Sec ion 6 concludes he pape .
2 Copula En opy
Wi h copula heo y, Ma and Sun [22] de ined he concep o Copula En opy as
ollows:
De ini ion 1 (Copula En opy).Le Xbe andom a iables wi h ma ginals u
and copula densi y unc ion c. The CE o Xis de ined as
Hc(x) = −Zu
c(u) log c(u)du.(1)
They also p oposed a non-pa ame ic es ima o o CE [22] comp ising o wo
simple s eps:
1. es ima ing empi ical copula densi y unc ion wi h ank s a is ic;
2. es ima ing he en opy o he es ima ed empi ical copula densi y wi h he
kNN en opy es ima o [27].
I he copula densi y unc ion cis gi en, he CE can also be es ima ed wi h
he ollowing way:
Hc(x) = −E(log c(u)).(2)
3 Tes ing Copula Hypo hesis
Gi en andom a iables X∈Rnand i s samples XTassocia ed wi h copula
densi y unc ion cx(u). Ou goal is o es whe he cbelong o a hypo hesis
2
c(u), he null hypo hesis o he p oblem is
H0:cx(u) = c(u); (3)
al e na i e hypo hesis is
H0:cx(u)=c(u).(4)
We p opose o es copula hypo hesis wi h CE. The p inciple o es ing is o
compa e he CE o he copula hypo hesis wi h he ue CE:
Tc(XT|c) = Hc(XT|c)−Hc(XT|cx),(5)
The i s e m is he CE unde he hypo hesis o copula cand he second e m
is ue CE. I H0is ue, hen Tcshould be 0; o he wise, Tcshould be la ge
alue.
Since CE is a uni ied heo y o copula unc ion, es ing copula hypo hesis
based on CE can be used o any ypes o copula unc ion. The only wo k
needed is o selec he copula amily c.
4 Es ima ion
The s a is ic in (5) can be es ima ed as wo pa . The second e m is ue
CE and he e o e can be es ima ed di ec ly om da a wi h he nonpa ame ic
es ima o o CE. The i s e m is he CE o copula hypo hesis which can be
es ima ed in he ollowing 3 s eps:
1. es ima e empi ical copula densi y ˆu om XT;
2. es ima e he pa ame e s αo copula cwi h ˆu;
3. calcula e he CE o he copula hypo hesis wi h he ollowing equa ion:
Hc(XT|c) = −E(log c(ˆu;α)).(6)
In he i s s ep, empi ical copula densi y can be es ima ed wi h ank s a is ic
and in he second s ep, he pa ame e s αo copula densi y can be es ima ed
wi h he likelihood me hod [2].
He e we gi e wo special cases o es ima ing CE o copula hypo hesis:
Gaussian Copula Gaussian copula densi y unc ion can be w i en as he
ollowing [28, 29]:
cn(u) = |Σρ|−
1
2exp −1
2Φ(u)(Σ−1
ρ−I)ΦT(u),(7)
whe e Σρis co ela ion ma ix, Φ is quan ile no mal unc ion, and Iis iden i y
ma ix.
So es ima ing CE o Gaussian copula hypo hesis can be done as ollows:
1. es ima e Σρ om XT;
2. calcula e he alue o Gaussian copula wi h 7;
3. calcula e CE o copula hypo hesis wi h 6.
3
A chimedean Copula Gumbel copula, F ank copula, and Clay on copula
a e h ee main membe s o A chimedean copula amily, and bi a ia e Gumbel
copula densi y unc ion is
cg(u) = exp
−"2
X
i=1
(−ln ui)α#
1
α
"2
X
i=1
(−ln ui)α#
1
α
−1
2
X
i=1
(−ln ui)α
ui!
,
(8)
bi a ia e F ank copula densi y unc ion is
c (u) = α(1 −e−α)e−α(u1+u2)
{1−e−α−(1 −e−αu1)(1 −e−αu2)}2,(9)
and bi a ia e Clay on copula densi y unc ion is
cc(u) = (α+ 1)(u1u2)−α−1(u−α
1+u−α
2+ 1)−2,(10)
whe e αis he pa ame e o Gumbel copula.
So es ima ing CE o A chimedean copula hypo hesis can be done as ollows:
1. es ima e αwi h he likelihood me hod [2];
2. calcula e he alue o A chimedean copula wi h (8), (9) o (10);
3. calcula e CE o Gumbel copula wi h (6).
5 Simula ions
We es he p oposed me hod wi h wo simula ion expe imen s 1. The i s
expe imen simula e bi a ia e Gaussian copula wi h co ela ion coe icien ρ
ange om 0.1 o 0.9 by he s ep 0.1. The second expe imen simula e bi a ia e
Gumbel, F ank, and Clay on copula wi h he pa ame e αchanging om 2
o 10 unde he ma gins being s anda d no mal dis ibu ion and exponen ial
dis ibu ion. All he sample size o simula ions is 300.
We applied he abo e es ing me hod o Gaussian copula hypo hesis and
A chimedean copula hypo heses o simula ed da a and de i ed ou es ima ed
s a is ics om each sample se .
The expe imen al esul s is shown in Figu e 1, Figu e 2, Figu e 3, and Figu e
4. I can be lea ned ha in he i s simula ion expe imen , he es ima ed
s a is ics o Gaussian copula hypo hesis is smalle han hose o A chimedean
copula hypo heses and ha in he second simula ion expe imen , he es ima ed
s a is ics o Gumbel, F ank, o Clay on copula hypo hesis is smalle han hose
o Gaussian copula hypo hesis and he o he A chimedean copulas. These mean
ha Gaussian copula hypo hesis is ue and Gumbel, F ank, o Clay on copula
hypo hesis is ue in wo expe imen s espec i ely.
The Rpackage copula [30] was used o A chimedean copula. The Rpackage
m no m [31] waw used o simula ing Gaussian dis ibu ion. The Rpackage
copen [32] was used as he implemen a ion o CE es ima o .
1The code is a ailable a h ps://gi hub.com/majian hu/ ch
4
0.2 0.4 0.6 0.8
−0.2 −0.1 0.0 0.1 0.2 0.3
Gaussian
ρ
s a is ic
Gaussian
Gumbel
F ank
Clay on
Figu e 1: Resul s o Gaussian copula hypo hesis simula ion expe imen s.
2 4 6 8 10
0123456
Gumbel
α
s a is ic
Gaussian
Gumbel
F ank
Clay on
Figu e 2: Resul s o Gumbel copula hypo hesis simula ion expe imen s.
5
2 4 6 8 10
0.0 0.5 1.0 1.5
F ank
α
s a is ic
Gaussian
Gumbel
F ank
Clay on
Figu e 3: Resul s o F ank copula hypo hesis simula ion expe imen s.
2 4 6 8 10
012345
Clay on
α
s a is ic
Gaussian
Gumbel
F ank
Clay on
Figu e 4: Resul s o Clay on copula hypo hesis simula ion expe imen s.
6
6 Conclusions
In his pape , we p oposed a copula hypo hesis es ing wi h copula en opy.
The es s a is ic is de ined. The es ima ion me hod o he p oposed s a is ic
is p oposed and wo special cases o es s o Gaussian copula hypo hesis and
A chimedean copula hypo hesis a e gi en. The e ec i eness o he p oposed
me hod is e i ied wi h simula ion expe imen s.
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