brief intro to copulas

8/12/2019 Brief Intro to Copulas

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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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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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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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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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(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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(0, 0)

(1, 1)

a u


Ca u , v =Ca av ,v =av

hen

11a vuav

C t ti C l


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(0, 0)

(1, 1)

a u


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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brief intro to copulas

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