Copulas: An Introduction Part II: Models

Copulas: An Introduction

Part II: Models

Johan Segers

Universit¨¦ catholique de Louvain (BE)

Institut de statistique, biostatistique et sciences actuarielles

Columbia University, New York City

9¨C11 Oct 2013

Johan Segers (UCL)

Copulas. II - Models

Columbia University, Oct 2013

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Copulas: An Introduction

Part II: Models

Archimedean copulas

Extreme-value copulas

Elliptical copulas

Vines

Johan Segers (UCL)

Copulas. II - Models

Columbia University, Oct 2013

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Copulas: An Introduction

Part II: Models

Archimedean copulas

Extreme-value copulas

Elliptical copulas

Vines

Johan Segers (UCL)

Copulas. II - Models

Columbia University, Oct 2013

3 / 65

The (in)famous Archimedean copulas

I

I

By far the most popular (theory & practice) class of copulas

Plenty of parametric models

I

I

Building block for more complicated constructions:

I

I

I

I

I

Gumbel, Clayton, Frank, Joe, Ali¨CMikhail¨CHaq, . . .

Nested/Hierarchical Archimedean copulas

Vine copulas

Archimax copulas

...

Mindless application of (Archimedean) copulas has drawn many

criticisms on the copula ¡®hype¡¯

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Columbia University, Oct 2013

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Laplace transform of a positive random variable

Recall the Laplace transform of a random variable Z > 0:

Z ¡Þ

¦×(s) = E[exp(?sZ)] =

e?sz dFZ (z),

s ¡Ê [0, ¡Þ]

0

A distribution on (0, ¡Þ) is identified by its Laplace transform.

Ex. Show the following properties:

I

I

I

I

0 ¡Ü ¦×(s) ¡Ü 1

¦×(0) = 1 and ¦×(¡Þ) = 0.

(?1)k dk ¦×(s)/dsk ¡Ý 0 for all integer k ¡Ý 1.

In particular, ¦× is nonincreasing (k = 1) and convex (k = 2).

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Copulas. II - Models

Columbia University, Oct 2013

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