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A Brief Introductionto Copulas
Speaker: Hua, LeiFebruary 24, 2009
Department of StatisticsUniversity of Britis !o"umbia
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Outline
#ntro$uctionDefinition%roperties&rcime$ean !opu"as
!onstructin' !opu"as(eference
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Introduction
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Introduction
)e *or$ Copulais a Latinnoun tat means ++& "ink, tie,
bon$++
!asse""+s Latin Dictionary-
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Introduction history
1959: The word Copulaappeared for thefirst time (Sklar 1959)
1981: The earliest paper relating copulas
to the study of dependence among randomvariales (Schwei!er and "olff 1981)
199#$s: %opula ooster: &oe (199') andelson (1999)
199#$s *: +cademic literatures on how touse copulas in risk management
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Introduction Why copula
on,linear dependence -e ale to measure dependence for heavy
tail distriutions .ery fle/ile: parametric0 semi,parametric
or non,parametric -e ale to study asymptotic properties of
dependence structures %omputation is faster and stale with the
two,stage estimation %an e more proailistic or more
statistical others
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Introduction Why copula
Example: X ~ lognormal(0, 1) and Y ~ lognormal(0, sigma^2)
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Introduction
Joint distribution funtion
Hx , y=P[Xx , Yy ]
!arginal distribution funtions
Fx =P[Xx ], G y =P[Yy ]
For each pair (x, y), we can associatethree numbers: F(x), G(y) and H(x, y)
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(x, y)
(1, 1)
(0, 0) F(x)
G(y)
Each pair of real number (x, y) leads to a point of(F(x), G(y)) in the unit square [0, 1![0, 1
H(x, y)
Introduction
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Introduction
)e mappin', .ic assi'ns te va"ueof te /oint $istribution function to eacor$ere$ pair of va"ues of mar'ina"
$istribution function is in$ee$ a copu"a
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Introduction
(x, y)
(1, 1)
(0, 0) F(x)
G(y)
H(x, y)Copulas
Joint distribution function
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"efinition
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Definition informal
& 2-dimensional copulais a$istribution function on 10, 310, 3,.it stan$ar$ uniform mar'ina"$istributions
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Definition a generic example
#f X, Y - is a pair of continuous ran$omvariab"es .it $istribution function Hx, y- an$mar'ina" $istributions Fxx- an$ FYy-respective"y, ten U 5 FXx- ~ U0, - an$ V 5
FYy- 6U0, - an$ te $istribution function ofU, V - is a copu"a
Cu ,v =PUu ,Vv=PXFX
1u ,YF
Y
1v
Cu ,v =HFX
1u, F
Y
1v
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Definition formal
Cu ,0=C0, v=0
Cu ,1=u C1,v =v
Cu2, v2Cu1, v 2Cu2, v1Cu1, v10
v1, v2, u1, u2[0,1] ; u2u1, v2v1
(u2, 2)
(u1, 1)
2"#nreasing
Grounded1$
2$
%$
C :[0,1 ]2
[0,1 ]
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#roperties
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Properties
$olume of %ectan&le
VH=Hu2, v2Hu1, v2Hu2, v1Hu1, v1
(u2, 2)
(u1, 1)
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Properties
'opula is the '$olume of rectan&le[0,u![0,
Cu , v =Vc[0, u ][0, v ]'opula assi&ns a number to each rectan&lein [0,1![0,1, *hich is nonne&atie +
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Properties
78amp"e: #n$epen$ent !opu"a
Cu1
, u2
=u1
u2
,u[0,1]2
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*+e grap+ of #ndependent opula
0002
0409
0:0
00
02
04
09
0:
0
00
02
04
09
0:
0
u
u2
!#
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Properties
Frchet !pper bound "opu#a
CUu1, u2=min {u1, u2 },u[0,1]2
Frchet $ower bound "opu#a
CLu1, u2=max {0,u1u21 },u[0,1]2
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Properties
Frchet $ower bound "opu#a Frchet !pper bound "opu#a
0002
0409
0:0
00
02
04
09
0:
0
00
02
04
09
0:
0
u
u2
!L
0002
0409
0:0
00
02
04
09
0:
0
00
02
04
09
0:
0
u
u2
!U
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Properties
%ny copu#a wi## be bounded by Frchet#ower and upper bound copu#as
CLu
1
, u2Cu
1
, u2C
Uu
1
, u2,u[0,1 ]2
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Properties
0002
0409
0:0
00
02
04
09
0:
0
00
02
04
09
0:
0
u
u2
!U#
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Properties
&'#ars heorem
Let H be a /oint $f .it mar'ina" $fs Fan$ G,)en tere e8ists a copu"a C suc tat
Hu , v =CFu, G v
If F and G are continuous,then the copula isunique
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Properties
*mportant "onse+uences & copu"a $escribes o. mar'ina"s are tie$
to'eter
& /oint $f can be $ecompose$ into mar'ina"$fs an$ copu"a mar'ina" $fs an$ copu"a can be stu$ie$
separate"y e': ;L7 separate"y-
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Properties
Surviva" copu"a Functiona" #nvariance for monotone transform
=on>parametric measures of $epen$ence )ai" $epen$ence Simu"ation
ther topics
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rchimedean 'opulas
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Archimedean Copulas
Cu , v =g[1 ]gugv
continuous, strict"y $ecreasin'conve8 function
g :[0,1 ][0,] g1=0
g[1]t={0, g0t
g1 t, 0tg0
!eudo"inver!eof g
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Archimedean Copulas
"ommutatie:
%ssociatie:
rchimedean 'opula behaes li-e a binary
operation
Cu , v =Cv , u, u , v[0,1]
CCu , v , # =Cu , Cv , #,u , v , #[0,1 ]
rder preserin-:
Cu1, v1Cu2, v2, u1u2, v1v2,[0,1 ]
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Archimedean Copulas
Example.
$et g t=1t , t[0,1 ]
g[1]
t=max1t ,0
Cu , v =max uv1,0
hen
Frchet $ower bound "opu#a is a 'ind of%rchimedean "opu#a.
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Archimedean Copulas
rchimedean 'opulas hae a *ide ran&e ofapplications for some reasons.
Easy to be constructed
/any families of copulasbelon& to it /any nice properties
rchimedean 'opulas ori&inally appeared in thestudy of probabilistic metric space, deelopin& theprobabilistic ersion of trian&le inequality
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he Inerse /ethod Geometric /ethods
'onstructin& 'opulas
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Constructing Copulas
he *nerse /ethod
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Constructing Copulas
he *nerse /ethod (xamp#e)Gumbel's bivariate exponential distribution
Hax , y ={
0, ot$er#i!e
1exeyexyaxy, x , y0
F1 u=ln 1u
G1 v=ln 1v
Ca u , v =uv11u1v ea ln 1u ln 1v
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Constructing Copulas
Geometric /ethods
2ithout reference to distribution
functions or random ariables, *e canobtain the copula ia the '$olume ofrectan&les in [0, 1![0, 1
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Constructing Copulas
Geometric /ethods (xamp#e)
(0, 0)
(1, 1)
a
let Cadenote the copula*ith supportas the linese&ments illustrated in the&raph
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Constructing Copulas
(0, 0)
(1, 1)
au
Geometric /ethods (xamp#e)continuous
Cau ,v =V
Ca
[0,u ][0,1 ]=u
uavhen
C i C l
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Constructing Copulas
(0, 0)
(1, 1)
a u
Geometric /ethods (xamp#e)continuous
Ca u , v =Ca av ,v =av
hen
11a vuav
C t ti C l
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Constructing Copulas
(0, 0)
(1, 1)
a u
Geometric /ethods (xamp#e)continuous
u11av
Ca u , v =uv1
VC
a
A=0 3
% hen
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%eference
R f
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Reference
?oe, H 99@- ;u"tivariate ;o$e"s an$ Depen$ence
!oncepts !apman A Ha""
2 =e"sen, (B 999-, &n #ntro$uction to !opu"as
Sc.eiCer, B an$ *o"ff, 7F 9- n nonparametric
measures of $epen$ence for ran$om variab"es &nn Statist9:@9>E
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