[PPT] - Generalized Diagonal Band Copulae with Two-Sided Generating PowerPoint Presentation

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Generalized Diagonal Band Copulae with Two-Sided Generating Densities

"Presentation Short Course: Beyond Beta and Applications" November 20th, 2018, La Sapienza

Samuel Kotz and J. René van Dorp1

Faculty Web-Page: www.seas.gwu.edu/~dorpjr

1 Corresponding Author, Department of Engineering Management and Systems Engineering, The

George Washington University, Washington D.C., USA

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OUTLINE

1. INTRODUCTION
2. COPULA CONSTRUCTION
3. GENERALIZED DIAGONAL BAND EXAMPLES
4. SAMPLING PROCEDURE
5. ORDINAL MEASURES OF ASSOCIATION
6. COPULA PARAMETER ELICITATION
7. A VALUE OF INFORMATION EXAMPLE
8. SELECTED REFERENCES

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1. INTRODUCTION...

Copula

,

: such that \ ] \ µ KÐ † Ñß ] µ LÐ † Ñ

w w w w

Continuous random variables

, H( ):

KÐ † Ñ † Cumulative distribution functions - cdf's.

The mapping

is called the \ Ä \ œ KÐ\ Ñ Ê Ó

w w

\ µ Ò!ß " Uniform probability integral transformation e.g. Nelsen (1999).

Any bivariate joint distribution of

can be transformed to a bivariate Ð\ ß ] Ñ

w w

copula { } - Sklar (1959). Ð\ß ] Ñ œ KÐ\ Ñß LÐ] Ñ

w w

Thus, a bivariate copula is a bivariate distribution with uniform marginals.
As such, many authors studied copulae indirectly.
Gaussian and Student-t Copulae (of this construct) were studied explicitlyÞ

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1. INTRODUCTION...

Archimedean Copula

Genest and Mackay (1986) used

for copula an algabraic method construction.

,

with

:

: À Ð!ß "Ó Ä Ò!ß ∞Ñ Ð"Ñ œ ! a convex decreasing function The generator function.

They possess

: joint cdf and probability density function (pdf) GÖBß Cl Ð † Ñ× œ Ð"Ñ Ö ÐBÑ  ÐCÑ× ÐBÑ  ÐCÑ Ÿ ! ! : : : : : : 

"

elsewhere

ÖBß Cl Ð † Ñ× œ 

Ð#Ñ ÖGÐBß CÑ× ÐBÑ ÐCÑ Ò ÖGÐBß CÑ×Ó : : : : : '' ' ' '

$

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1. INTRODUCTION...

Diagonal Band Copula

Cooke and Waij (1986) used

for copula construction a geometric method

1 1 0.00 1.00 2.00 3.00 4.00

x y

(0, 1−θ) (1, θ) DB2 DB1 DB3

A

B

y

(0, θ−1) (1, 2−θ) (0, 1) (1, 0) (0, 0) (1, 1)

1 1 0.00 1.00 2.00 3.00 4.00

x y

(0, 1−θ) (1, θ) DB2 DB1 DB3

A

B

y

(0, θ−1) (1, 2−θ) (0, 1) (1, 0) (0, 0) (1, 1)

Figure 1: A: Gray area support of a copula comprised of sub-areas HFÐ Ñ ) HF ß 3 œ "ß #ß $à HFÐ!Þ&Ñ

3

B: Example of a copula.

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1. INTRODUCTION...

Diagonal Band Copula

Diagonal Band (DB) copula possess pdf:

GÖBß Cl × œ Ð$Ñ "ÎÐ"  Ñ ÐBß CÑ − HF ∪ HF "ÎÖ#Ð"  Ñ× ÐBß CÑ − HF ! ) ) )   

" $ #

elsewhere

,

Analagous to Archimedean copula Bojarski (2001) generalized copula via HFÐ Ñ ) a generator function | . 0Ð † Ñ )

Generator function

| is a with support . 0Ð † Ñ Ò Ó ) ) symmetric pdf  "ß "  )

Lewandowski (2005) showed that Bojarski's (2001) GDB Copulae are

equivalent to Fergusons (1995) family of copulae with joint pdf:

ÐBß CÑ œ

Ö1ÐlB  Cl  1Ð"  l"  B  ClÑ× Ð%Ñ " # , 1Ð † Ñ Ò!ß "Ó pdf on

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1. INTRODUCTION...

Generalized DB Copula

For sampling efficiency
f generator

would be desirable. inverse cdf 0Ð † l Ñ )

Consider Van Dorp and Kotz's (2003) symmetric Two-Sided (TS) pdf's À

0 Dl:Ð † l Ñ× œ ‚ " # :ÐD  "l Ñ  "  D Ÿ !ß :Ð"  Dl Ñ !  D  "ß { , for , for G G G  Ð&Ñ that too uses the generating pdf

concept. Pdf

has support :ÐDÑ :ÐDÑ Ò!ß "ÓÞ

The

associated with inverse cdf (or quantile function) Ð%Ñ J ?l:Ð † l Ñ× œ !  ? Ÿ ß  ?  "ß

" " # " #

{ , for , for G G G  T Ð#?l Ñ  " "  T Ð#  #?l Ñ Ð'Ñ

" "

where is the quantile function of T Ð † l Ñ

"

< :Ð † l ÑÞ G

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OUTLINE

1. INTRODUCTION
2. COPULA CONSTRUCTION
3. GENERALIZED DIAGONAL BAND EXAMPLES
4. SAMPLING PROCEDURE
5. ORDINAL MEASURES OF ASSOCIATION
6. COPULA PARAMETER ELICITATION
7. A VALUE OF INFORMATION EXAMPLE
8. SELECTED REFERENCES

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2. CONSTRUCTION...

GDB Copula with TS Gen. PDF

Bivariate pdf

is constructed where and 1ÐBß CÑ ß \ µ YÒ!ß "Ó the conditional pdf 1ÐClBÑ has the following form À 1ÖClBß × œ Ð(Ñ :Ð † l Ñ l:Ð † l Ñ× G G 0ÖB  C B  " Ÿ C Ÿ B  " , ,

From

, and it follows that: \ µ YÒ!ß "Ó Ð(Ñ TS framework pdf Ð%Ñ 1ÖBß Cl œ Ð)Ñ ß ß :Ð † l Ñ× ‚ " # :Ð"  l Ñ :Ð"  l Ñ G G G  B  C  "  B  C Ÿ !ß B  C !  B  C  "ß

From

, a bivariate pdf | is constucted on the unit square Ð)Ñ

ÐBß C

Ñ :Ð † l Ñ G Ò!ß "Ó 1ÖBß Cl

#

f
utside

by folding back the probability masses :Ð † l Ñ× G the unit square

nto it,

Ò!ß "Ó# using "folding" lines and . C œ " C œ !

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2. CONSTRUCTION...

GDB Copula with TS Gen. PDF

1 1 0.00 1.00 2.00

x y 1 1 x – y = – 1 x – y = 1

B

x – y = 0 (0,0) (1,1) (1,0) (0,1)

C

2

A

4

A

3

A

1

A

x + y = 1 x + y = 0 x + y = 2 x y

1

A

3

A

2

A

4

A

Figure 1. A: pdf B: Areas 1ÐBß CÑ Ð)Ñà E ß 3 œ "ß á ß %à

3

C: pdf with

n

.

ÖBß Cl

× Ð"!Ñ :ÐDÑ œ #D Ò!ß "Ó :Ð † l Ñ G

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2. CONSTRUCTION...

GDB Copula with TS Gen. PDF

Relationship between

and

ÖBß Cl

× 1ÖBß Cl :Ð † l Ñ :Ð † l Ñ× G G in 8 Ð Ñ À

ÖBß Cl

× œ 1ÖBß Cl  1ÖBß  Cl ß 1ÖBß Cl  1ÖBß #  Cl ß :Ð † l Ñ Ð*Ñ :Ð † l Ñ× :Ð † l Ñ× :Ð † l Ñ× :Ð † l Ñ× G G G G G  !  B  C Ÿ " "  B  C Ÿ # , .

Combining

with Ð*Ñ Ð)Ñ now yields À

ÖBß Cl

× œ " # ‚ "  ß "  ß ß :Ð † l Ñ :Ð l Ñ  :Ð"  l Ñ :Ð l Ñ  :Ð"  l Ñ :Ð l Ñ  :Ð"  l Ñ :Ð l Ñ  G G G G G G G G        B  C B  C ÐBß CÑ − E B  C B  C ÐBß CÑ − E B  C  " B  C ÐBß CÑ − E B  C  "

" # $

, , , :Ð"  l Ñ B  C ÐBß CÑ − E Þ G ß

%

Ð"!Ñ

Note in Ð"!Ñ -ÐCß B œ -ÐBß CÑ

\ µ Y Ò!ß "Ó Ê ] µ Y Ò!ß "Ó Ñ . Hence,

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2. CONSTRUCTION...

Joint CDF

Pdf of GDB copula with TS pdf with generating pdf :ÐDlGÑ À
ÖBß Cl

× œ " # ‚ "  ß "  ß ß :Ð † l Ñ :Ð l Ñ  :Ð"  l Ñ :Ð l Ñ  :Ð"  l Ñ :Ð l Ñ  :Ð"  l Ñ :Ð l Ñ  G G G G G G G G        B  C B  C ÐBß CÑ − E B  C B  C ÐBß CÑ − E B  C  " B  C ÐBß CÑ − E B  C  "

" # $

, , , :Ð"  l Ñ B  C ÐBß CÑ − E Þ G ß

%

Cdf of GDB copula with TS gen. pdf :ÐDlGÑ and cdf T ÐDl

Ñ G follows as: GÖBß Cl × œ  ß ß  ß :Ð † l Ñ B TÐDl Ñ.D C  TÐDl Ñ×.D B TÐDl Ñ.D C G G G G             

" # " " # " " #

  

BC B " BC BC # C" BC $ " C " B "

ÐBß CÑ − E ÐBß CÑ − E ÐBß CÑ − E , , ,  TÐDl Ñ.D

" #BC" BC % "

G ß Ð""Ñ ÐBß CÑ − E Þ

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OUTLINE

1. INTRODUCTION
2. COPULA CONSTRUCTION
3. GENERALIZED DIAGONAL BAND EXAMPLES
4. SAMPLING PROCEDURE
5. ORDINAL MEASURES OF ASSOCIATION
6. COPULA PARAMETER ELICITATION
7. A VALUE OF INFORMATION EXAMPLE
8. SELECTED REFERENCES

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3. GDB-TS EXAMPLES...

Triangular PDF

Substitution of generating pdf :ÐDÑ œ #D with support

in yields Ò!ß "Ó Ð"!Ñ

ÐBß CÑ œ # ‚

" ß "  ß ß ß   C ÐBß CÑ − E B ÐBß CÑ − E B ÐBß CÑ − E C ÐBß CÑ − E Þ

" # $ %

, , ,

1 1 0.00 1.00 2.00

A B 1.0 0.5 1.0 0.5 0.0 2.0 1.5 1.5 2.0 1.0 0.5 1.5 0.0 0.5 1.0 1.5

x y

1

A

3

A

2

A

4

A

1

A

3

A

2

A

4

A

Figure 2. A: Copula density ; B: Density contour plot

ÖBß C×

Þ

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3. GDB-TS EXAMPLES...

Triangular PDF

Substitution of pdf :ÐDÑ œ #D

Ð""Ñ in and yields: generating cdf T ÐDÑ œ D# GÖBß C× œ ‚ " $  B  $BC  'BC ß C  $B C  'BCß C  $C  $CÐB  "Ñ  $B  $B  "ß B  $B  $BÐC  "Ñ  $C  $C  "ß       

$ # $ # $ # # # $ # # #

ÐBß CÑ − E ÐBß CÑ − E ÐBß CÑ − E ÐBß CÑ − E

" # $ %

, , ,  Þ

1 1 0.00 0.20 0.40 0.60 0.80 1.00

x y

0.5 1 1.5 2 0.2 0.4 0.6 0.8 1

z Generating PDF of TS Fraemework

Figure 3. Graph of joint triangular copula cdf GÐBß CÑ given aboveÞ

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3. GDB-TS EXAMPLES...

Slope PDF

:ÐDl Ñ œ  #Ð Dß ! Ÿ Ÿ #ß α α #   "Ñ α α

1 1 0.00 1.00 2.00 3.00 4.00

0.5 1 1.5 2 2.5 3 3.5 4 0.2 0.4 0.6 0.8 1

z Slope PDF

A B

Figure 4. A: Slope generating pdf; B: GDB Copula with TS Gen. PDF in AÞ

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3. GDB-TS EXAMPLES...

Power PDF

:ÐDl8Ñ œ 8D ß 8  !ß

8"

1 1 0.00 1.00 2.00 3.00 4.00

0.5 1 1.5 2 2.5 3 3.5 4 0.2 0.4 0.6 0.8 1

z Power PDF

C D

Figure 5. C: Power generating pdf; D: GDB Copula with TS Gen. PDF in AÞ

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3. GDB-TS EXAMPLES...

Ogive PDF

:ÐDl7Ñ œ Ö# 7  " D  7D ×ß 7  ! 7  # $7  % ( ) . 

7 7"

1 1 0.00 1.00 2.00 3.00 4.00

0.5 1 1.5 2 2.5 3 3.5 4 0.2 0.4 0.6 0.8 1

z Ogive PDF

E F

Figure 6. E: Ogive generating pdf; F: GDB Copula with TS Gen. PDF in AÞ

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3. GDB-TS EXAMPLES...

Uniform PDF Ò ß "Ó )

:ÐDl Ñ œ ß Ÿ D Ÿ "ß ! Ÿ Ÿ "ß " "  ) ) ) )

1 1 0.00 1.00 2.00 3.00 4.00

0.5 1 1.5 2 2.5 3 3.5 4 0.2 0.4 0.6 0.8 1

z Uniform PDF

G H

Figure 7. G: Uniform gen pdf; H: Ò ß "Ó Þ )

GDB Copula with TS Gen. PDF in AÞ

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3. GDB-TS EXAMPLES...

Beta PDF

:ÐDl+ß ,Ñ œ B Ð"  BÑ Ð+  ,Ñ Ð+Ñ Ð,Ñ > > >

+" ,",

, +  !ß ,  !

1 1 0.00 1.00 2.00 3.00 4.00

0.5 1 1.5 2 2.5 3 3.5 4 0.2 0.4 0.6 0.8 1

z Slope PDF

I J

Figure 8. G: Beta generating pdf; H: GDB Copula with TS Gen. PDF in AÞ

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OUTLINE

1. INTRODUCTION
2. COPULA CONSTRUCTION
3. GENERALIZED DIAGONAL BAND EXAMPLES
4. SAMPLING PROCEDURE
5. ORDINAL MEASURES OF ASSOCIATION
6. COPULA PARAMETER ELICITATION
7. A VALUE OF INFORMATION EXAMPLE
8. SELECTED REFERENCES

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4. SAMPLING PROCEDURE...

Algorithm

x y 1 y = 1

1. Sample x in [0,1]
2. Sample z in [-1,1]
3. y = z + x
4. If y < 0 then y = −

= −y

5. If y > 1 Then y =1

=1−( −( y−1) −1) ALGORITHM: z x y = = x y y y = −1

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4. SAMPLING PROCEDURE...

Algorithm

Thus, sampling algorithm mimics construction method À

Step 1: Sample from a uniform random variable

n

. B \ Ò!ß "Ó Step 2: Sample from a uniform random variable

n

. ? Y Ò!ß "Ó Step 3: If then else ? Ÿ D œ T Ð#?Ñ  " D œ "  T Ð#  #

" # " "

?Ñ Step : % C œ D  B Step 5: If then C  ! C œ  C Step 6: If then C  " C œ "  ÐC  "Ñ

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4. SAMPLING PROCEDURE...

Algorithm

For the generating densities herein we have for arbitrary quantile level

; − Ð!ß "Ñ: T ß :ÐDl Ñß Á "ß ; ß :ÐDl8Ñß   ; ß :ÐDl7Ñß Ð"  Ñ;  ß

" Ð# Ñ Ð# Ñ %Ð "Ñ; #Ð "Ñ "Î8 #Ð7"Ñ #Ð7"Ñ 7 7 7 # $7% #ÎÐ7#Ñ

Ð;l Ñ œ <                   

α α α α 

#

α α ) ) :ÐDl Ñß )

One could favor the power pdf and uniform pdf's due to least number of
perations.

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OUTLINE

1. INTRODUCTION
2. COPULA CONSTRUCTION
3. GENERALIZED DIAGONAL BAND EXAMPLES
4. SAMPLING PROCEDURE
5. ORDINAL MEASURES OF ASSOCIATION
6. COPULA PARAMETER ELICITATION
7. A VALUE OF INFORMATION EXAMPLE
8. SELECTED REFERENCES

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5. ORDINAL MEASURES OF ASSOCIATION...

Overview

Positive (negative) dependence between

and \ µ KÐ † Ñ ] µ LÐ † Ñ

w w

when

f one go with
f the other.

large values large (small) values

In case of positive (negative) dependence,

and are said to be \ ]

w w

concordant (disconcordant).

Classical measures for

: the degree of positive or negative dependence Blomquist's (1950) , Kendall's (1938) and Spearman's (1904) . " 7 3=

All three measures attain

. values ranging from to  " "

All three are

Hence,

rdinally invariantÞ

3 3

= = w w w w

Ð\ ß ] Ñ œ Ð\ß ] Ñß Ð\ß ] Ñ œ ÖKÐ\ Ñß LÐ] Ñ× where , etc.

Recall,

is a copula. \ ] µ YÒ!ß "Ó and and thus the joint pdf of Ð\ß ] Ñ

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5. ORDINAL MEASURES OF ASSOC. ...

Blomquist's "

Excellent review of classical measures , and

is given by Kruskal (1958). " 7 3= "Ð\ß ] Ñ œ %GÐ ß Ñ  " " " # # , where is copula cdf GÐ † ß † Ñ X1 < 0.5 X1 > 0.5 Y1 < 0.5 Y1 > 0.5 1

1

Y1 < 0.5 Y1 > 0.5 1

1

EMV = β A Blomqvist’s β

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5. ORDINAL MEASURES OF ASSOC. ...

Kendalls's 7

Let -Ð † ß † Ñ GÐ † ß † Ñ

\ ß ] Ñ µ GÐ † ß † Ñ , be the copula pdf and cdf , and let Ð

3 3

3 œ "ß # be two independent bivariate samples from the copula. 7Ð\ß ] Ñ œ % GÐBß CÑ-ÐBß CÑ.B.C  "  

! ! " "

. X1 < X2 X1 > X2 Y1 < Y2 Y1 > Y2 1

1

Y1 < Y2 Y1 > Y2 1

1

EMV = τ B Kendall’s τ

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5. ORDINAL MEASURES OF ASSOC. ...

Spearman's 3=

Let -Ð † ß † Ñ GÐ † ß † Ñ

\ ß ] Ñ µ GÐ † ß † Ñ 3 œ "ß #ß $ , be the copula pdf , let Ð

3 3

be three independent bivariate samples from the copula. 3=

! ! " "

Ð\ß ] Ñ œ "# BC-ÐBß CÑ.B.C  $   Þ X1 < X2 X1 > X2 Y1 < Y3 Y1 > Y3 1

1

Y1 < Y3 Y1 > Y3 1

1

EMV = ρs C Spearman’s ρs

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5. ORDINAL MEASURES OF ASSOC. ...

GDB-TS Copula

Summarizing, population expressions for , and

are " 7 3= :          " 7 3 Ð\ß ] Ñ œ %GÐ ß Ñ  " Ð\ß ] Ñ œ % GÐBß CÑ-ÐBß CÑ.B.C  " Ð\ß ] Ñ œ "# BC-ÐBß CÑ.B.C  $

" " # # ! ! " " = ! ! " "

, , ,

We have for GDB copula with TS pdf with generating pdf

and :Ð † l Ñ G ^ µ :Ð † l Ñ G :      " G G 7 G G 3 G G G Ö\ß ] l:Ð † l Ñ× œ #IÒ^l Ó  " Ö\ß ] l:Ð † l Ñ× œ l Ö\ß ] l:Ð † l Ñ× œ  , , . #IÒ^ Ó = .=  % = = .=  "

# ! ! " "

 # T Ð l Ñ T Ð l Ñ  

# #

G G

=

% l Ó  'IÒ^ l Ó  " IÒ^$

#

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5. ORDINAL MEASURES OF ASSOC. ...

GDB-TS Copula

Slope pdf : :ÐDl Ñ œ

 #Ð Dß ! Ÿ Ÿ #ß α α #   "Ñ α α      " α α 7 α 3 α α Ö\ ] × œ Ö\ ] × œ  Ö\ ] œ , , , l l l :Ð † l Ñ   ß − Ò  ß Óß :Ð † l Ñ  ß − Ò  ß Óß :Ð † l Ñ  ß − Ò  ß Ó

" " " " $ $ $ $ % % % "& "& "& = # # & & % "& # # & &

α  .

Power pdf :

:ÐDl8Ñ œ 8D ß 8  !ß

8"

       " 7 3 Ö\ ] × œ Ö\ ] × œ Ö\ ] × œ , , , l l l :Ð † l8Ñ ß − Ò  "ß "Óß :Ð † l8Ñ  ß − Ò  "ß "Óß :Ð † l8Ñ ß − Ò  "

8" 8" 8" 8" 8# Ð8"ÑÐ8#ÑÐ#8"Ñ = Ð8"ÑÐ8'Ñ Ð8#ÑÐ8$Ñ

ß "Ó.

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5. ORDINAL MEASURES OF ASSOC. ...

GDB-TS Copula

Ogive pdf: :ÐDl7Ñ œ

Ö# 7  " D  7D ×ß 7  !ß

7# $7% 7 7"

( )        " 7 Ö\ ] × œ Ö\ ] × œ , , l l :Ð † l7Ñ ß − Ò!ß "Óß :Ð † l7Ñ

7Ð7"ÑÐ$7)Ñ Ð7 ÑÐ7 ÑÐ$7%Ñ 3 4 7Ð7"ÑÐ"'#7 #'%$7 ")"$#7 ''"!)7 "%!!$#7&)))!Ñ Ð7$Ñ

' & % $

Ð7%ÑÐ7'ÑÐ#7&ÑÐ$7%Ñ Ð$7)ÑÐ$7"!Ñ

#

ß − Ò!ß "Óß :Ð † l7Ñ ß − Ò!ß "Ó 3=

7Ð7"ÑÐ$7 (!7 %#%7($'Ñ Ð7%ÑÐ7&ÑÐ7'ÑÐ7)ÑÐ$7%Ñ

Ö\ ] × œ , l

$ #

.

U

pdf Ò ß "Ó À ) :ÐDl Ñ œ ß Ÿ D Ÿ "ß ! Ÿ Ÿ " " "  ) ) ) ) ß    " ) ) 7 ) ) 3 ) Ö\ ] × œ Ö\ ] × œ Ö\ ] × œ , , , l l Ð  # l Ð"   Ñ :Ð † l Ñ ß − Ò!ß "Óß :Ð † l Ñ ÑÎ$ß − Ò!ß "Óß :Ð † l Ñ ß − Ò!ß "Ó ) ) ) )

= #

.

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5. ORDINAL MEASURES OF ASSOC. ...

GDB-TS Copula

0.00 0.25 0.50 0.75 1.00 5 10 15 20 m

0.50
0.25

0.00 0.25 0.50 0.5 1 1.5 2 α

1.00
0.75
0.50
0.25

0.00 0.25 0.50 0.75 1.00 1 2 3 4 5 6 7 8 9 10 n 0.00 0.25 0.50 0.75 1.00 0.25 0.5 0.75 1 θ

A B D

β τ ρs ρs

C

Figure 8. A: Slope( ) B: Power( C: Uniform ; D: Ogive( . α ) à 8Ñà Ò ß "Ó 7Ñ

SLIDE 34

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5. ORDINAL MEASURES OF ASSOC. ... Reflection Property
Let

be pdf , ;ÐDl Ñ ^ œ "  ^ ^ µ :ÐDl Ñ Ê G G

w

;ÐDl Ñ œ :Ð"  Dl Ñ G G .

ÖBß Cl;

× œ -ÖBß Cl × ÐDl Ñ :Ð"  Dl Ñ G G

btained via a right angle rotation.

1 1 0.00 1.00 2.00

x y

Figure 9. Graph of rotated copula . using :Ð"  Dl Ñ œ #Ð"  DÑ G

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5. ORDINAL MEASURES OF ASSOC. ... Reflection Property
We have for GDB copula with TS gen. pdf

and : :Ð † l Ñ ^ µ :Ð † l Ñ G G      " G G 7 G 3 G G G Ö\ß ] l:Ð † l Ñ× œ #IÒ^l Ó  " Ö\ß ] l:Ð † l Ñ× œ Ö\ß ] l:Ð † l Ñ× œ  , , . #IÒ^ Ó = .=  % = = .=  "

# ! ! " "

 # T Ð l Ñ T Ð l Ñ  

# #

G G

=

% l Ó  'IÒ^ l Ó  " IÒ^$

#

Let

be pdf , ;ÐDl Ñ ^ œ "  ^ ^ µ :ÐDl Ñ G G

w

Ê    " " " 7 7 7 3 3 3 Ö\ß ] l × œ Ö\ß ] l × œ  Ö\ß ] l × Ö\ß ] l × œ Ö\ß ] l × œ  Ö\ß ] l × Ö\ß ] l × œ Ö\ß ] l × œ  ;ÐDl Ñ :Ð"  Dl Ñ :ÐDl Ñ ;ÐDl Ñ :Ð"  Dl Ñ :ÐDl Ñ ;ÐDl Ñ :Ð"  Dl Ñ G G G G G G G G , ,

= = =Ö\ß ] l

× :ÐDl Ñ G .

n

:ÐDl Ñ Ê G symmetric Ò!ß "Ó Ê :Ð"  Dl Ñ G G œ :ÐDl Ñ ß ß ´ ! " 7 3=

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5. ORDINAL MEASURES OF ASSOC. ... Reflection Property

:ÐDl+Ñ œ B Ð"  BÑ Ð#+Ñ Ð+Ñ Ð+Ñ > > >

+" +", +  ! Ê " 7 3

ß ß ´ ! a +  !

=

,

1 1 0.00 1.00 2.00 3.00 4.00

0.5 1 1.5 2 2.5 3 3.5 4 0.2 0.4 0.6 0.8 1

z Slope PDF

I J

Figure 10. G: Beta generating pdf; H: GDB Copula with TS Gen. PDF in AÞ

SLIDE 37

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5. ORDINAL MEASURES OF ASSOC. ...

Tail Dependence

Lower and upper tail dependence measures are in vogue, particularly in

problem contexts dealing with modeling the joint occurrence of extreme events, such as insurance and modeling of default risk in finance.

Recent

to the copula approach may be credited to burst of attention the Gaussian copula which has been widely adopted by the "financial quants" .

Embrechts (2008) even refers to this attention as "the copula craze".
Unfortunately, some (see, e.g.,

Gaussian copula for the Salmon, 2009) blamed 2008 financial crash lack of , in part due to lower and upper tail dependence.

These measures too are

, although they

rdinal measures of association

focus primarily on modeling positive dependence and not negative dependence.

SLIDE 38

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5. ORDINAL MEASURES OF ASSOC. ...

Tail Dependence

\ µ KÐ † Ñ ] µ LÐ † Ñ

' , ' , : Lower tail dependence -P

P

" "

œ T<Ö] Ÿ L ÐBÑl\ Ÿ K ÐBÑ× B Æ ! œ T<Ð] Ÿ Bl\ Ÿ BÑ œ ß B Æ ! B Æ ! GÐBß BÑ B lim lim lim ' '

\ µ KÐ † Ñ ] µ LÐ † Ñ

' , ' , Upper tail dependence -Y :

Y

" "

œ T<Ö]  L ÐBÑl\  K ÐBÑ× B Å " œ T<Ð]  Bl\  BÑ œ Þ B Å " B Å " "  #B  GÐBß BÑ "  B lim lim lim ' '

Clayton, Frank and Gumbel copulae exhibit lower or upper tail dependence.

Clayton, Frank and Gumbel copulae belong to the Archimedean class.

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5. ORDINAL MEASURES OF ASSOC. ...

Tail Dependence

For GDB copula cdf

we have , GÖBß Cl × :Ð † l Ñ G

P

Y

œ œ ! similar to the Gaussian copulae (Embrechts et. al, 2002).

Blomquist's

Kendall's and Spearman's are more applicable in contexts " 7 3 ß

W

dealing with the modeling of joint events in general, not extremes per se.

Blomquist's

Kendall's and Spearman's pertain to full copula support " 7 3 ß

W

and not just to their asymptotic extreme values.

Clayton, Frank and Gumbel copulae

Caution to those who believe that the could serve as the panacea instead of Gaussian Copula.

Heteroscedastic behavior of financial processes

their dependence suggests cannot be modelled using a copula with , a constant correlation over time regardless of the copula displaying tail dependence or not.

SLIDE 40

BEYOND BETA SHORT COURSE: La Sapienza J.R. van Dorp; dorpjr@gwu.edu - Page 39

OUTLINE

1. INTRODUCTION
2. COPULA CONSTRUCTION
3. GENERALIZED DIAGONAL BAND EXAMPLES
4. SAMPLING PROCEDURE
5. ORDINAL MEASURES OF ASSOCIATION
6. COPULA PARAMETER ELICITATION
7. A VALUE OF INFORMATION EXAMPLE
8. SELECTED REFERENCES

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6. ELICITATION... Procedure COPULA PARAMETER

\ µ KÐ † Ñß ] µ LÐ † Ñ ÖKÐ\ Ñß LÐ] Ñ× œ Ð\ß ] Ñ µ GÖBß Cl

× ' ' , ' ' :Ð † l Ñ G

Elicit: T <Ð

Ÿ l Ÿ Ñ œ T <Ð] Ÿ !Þ&l\ Ÿ !Þ&Ñ œ ] C \ B ' ' ' '

!Þ& !Þ&

1.

This

falls within elicitation procedure the conditional fractile estimation method for eliciting degree of dependence - Clemen and Reilly (1999).

We have for Blomquist's "

" G 1 G Ö\ß ] l:Ð † l Ñ× œ # Ö\ß ] l:Ð † l Ñ×  "

Hence, elicitation of 1

G Ö\ß ] l:Ð † l Ñ× is equivalent to an indirect elicitation

f Blomquist's " which has a more straightforward interpretation as and

7 3=Þ

SLIDE 42

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6. ELICITATION... Procedure COPULA PARAMETER

GÖBß Cl × œ  ß ß  ß :Ð † l Ñ B TÐDl Ñ.D C  TÐDl Ñ×.D B TÐDl Ñ.D C G G G G             

" # " " # " " #

  

BC B " BC BC # C" BC $ " C " B "

ÐBß CÑ − E ÐBß CÑ − E ÐBß CÑ − E , , ,  TÐDl Ñ.D

" #BC" BC % "

G ß ÐBß CÑ − E Þ Ê " # œ T <Ð] Ÿ !Þ&ß \ Ÿ !Þ&Ñ œ 1 GÖ ß l × œ " " # # :Ð † l Ñ Ö"  T ÐDl Ñ×.D " # G G 

! "

With ^ µ

IÒ^ Ó œ ß TÐDl Ñß G l Ö"  TÐDl Ñ×.D G G !

"

thus we have: 1 G Ö\ß ] l:Ð † l Ñ× œ IÒ^l Ó G Þ Ð"#Ñ

Thus, having elicited one solves for using

. 1 < the method of moments

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6. ELICITATION... Procedure COPULA PARAMETER

We have for the different pdf's herein

1 G α Ö\ ] × œ , l:Ð † l Ñ Ð − Ò ß Ó :ÐDl Ñ 8ÎÐ8  "Ñ − Ò!ß "Ó :ÐDl8Ñ − Ò!Þ&ß "Ó :ÐDl7            #  ÑÎ' α

" # $ $ Ð7#Ñ $7% Ð7%ÑÐ7$Ñ $7'

, , Slope pdf, , , Power pdf, ,

#

Ñß Ð  "ÑÎ# − Ò!Þ&ß "Ó :ÐDl Ñ Ò ß "Ó Ogive pdf, , , U pdf. ) ) )

Assume that an expert has assessed a value

1 G Ö\ ] × , l œ !Þ(& :Ð † l Ñ T<Ð] Ÿ !Þ&l\ Ÿ !Þ&Ñ œ .

How does one select a GDB copula with a TS gen. pdf that matches?

SLIDE 44

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6. ELICITATION... Smoothness? COPULA PARAMETER

1 1 0.00 1.00 2.00 3.00 4.00

Power

1 1 0.00 1.00 2.00 3.00 4.00

Ogive

1 1 0.00 1.00 2.00 3.00 4.00

U[θ,1]

All three match the constraint

1 G Ö\ ] × , l œ !Þ(& :Ð † l Ñ T<Ð] Ÿ !Þ&l\ Ÿ !Þ&Ñ œ

How do we select one? If smoothness is required

Ogive pdf. Ê

What if smoothness is not required? Pick one that is uniform as possible?

SLIDE 45

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6. ELICITATION... Smallest ? COPULA PARAMETER 3=

In figures above

, and subsitution in formulas: 8 œ $ß 7 œ %Þ*"' œ "Î# ) 3= 3 3 3 )

= = =

Ð8Ñ œ ß Ð7Ñ œ !Þ'!&* Ð Ñ œ Ð$)Ñ $ & & ) and .

Can we select the one with the smallest rank correlation in general?
Pdf :Ð † l Ñ

Ò!ß "Ó Ê IÒ^l Ó œ :Ð † l Ñ G < 1 G symmetric on Ö\ ] × œ Þ , l

" #

Pdf :Ð † l Ñ

Ò!ß "Ó Ê :Ð † l Ñ× ´ !Þ G 3 G symmetric on

=Ö\ ]

, l

When elicited 1

G Ö\ ] × œ œ , it seems to l:Ð † l Ñ T<Ð] Ÿ !Þ&l\ Ÿ !Þ&Ñ

" #

intuitive to select copula with independent uniform marginals.

Hence, answer is: No.

SLIDE 46

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6. ELICITATION...

Max. Entropy?

COPULA PARAMETER

Select GDB copula that

between it and copula with minimizes distance independent uniform marginals.

Kullback-Liebler distance

the relative information measures

f one

candidate pdf with respect to pdf given by 0ÐBß CÑ 1ÐBß CÑ MÐ0l1Ñ œ 0ÐBß CÑ Ö0ÐBß CÑÎ1ÐBß CÑ×.B.C   ln .

Setting

and yields: 0ÐBß CÑ œ -ÖBß C× 1ÐBß CÑ œ ?ÐBß CÑ œ "ß MÐ-l?Ñ œ

ÐBß CÑ Ö-ÐBß CÑ×.B.C

Ð%!Ñ  

W-

ln ,

The quantity

is known as

f the pdf

I œ  MÐ-l?Ñ

ÐBß CÑÞ

the entropy

Bedford and Meeuwissen (1997) constructed maximum entropy copulae

given a correlation constraint that are . minimally informative

SLIDE 47

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6. ELICITATION...

Max. Entropy

COPULA PARAMETER

Bedford and Meeuwissen's (1997) maximum entropy copulae given a

correlation constraint, do not possess closed form pdf and cdf.

Select amongst a set of GDB copula that matches a specified constraint that
ne that is

(or has maximum entropy). minimally informative

Utilizing
ver a 100 by 100 grid over

we have numerical integration Ò!ß "Ó ß

#

MÖ-ÐBß CÑl:Ð † l Ñ× œ !Þ#"$' :ÐDl8Ñß 8 œ $ß !Þ#### :ÐDl7Ñß 7 œ %Þ*"' !Þ$%!! :ÐDl Ñß œ !Þ& Ò ß "Ó < ) ) )    , Power pdf, , , Ogive pdf , . U pdf

Summarizing, given

set by the constraint 1 œ !Þ(& T<Ð] Ÿ !Þ&l\ Ÿ !Þ&Ñ œ , the relative information approach above would suggest to use the GDB copula with the power( ) generating pdf with 8 Ð#'Ñ 8 œ $.

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6. ELICITATION...

Max. Entropy

COPULA PARAMETER

0.0 0.5 1.0 1.5 2.0 0.4 0.5 0.6 0.7 0.8 0.9 1

ρ

Relative Information

DB POWER OGIVE

Min. Rel. Info.

0.0 0.5 1.0 1.5 2.0 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

π Relative Information

DB POWER OGIVE 0.00 0.05 0.10 0.15 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

ρ Relative Information

DB SLOPE POWER OGIVE

Min. Rel . Info.

0.00 0.05 0.10 0.15 0.5 0.55 0.6 0.65

π Relative Information

DB SLOPE POWER OGIVE

A D C B

A: B: : D: . !Þ& Ÿ Ÿ à ! Ÿ Ÿ !Þ%à G Ÿ Ÿ !Þ*&à ! % Ÿ Ÿ !Þ** 1 3 1 3

# # $ $

SLIDE 49

BEYOND BETA SHORT COURSE: La Sapienza J.R. van Dorp; dorpjr@gwu.edu - Page 48

OUTLINE

1. INTRODUCTION
2. COPULA CONSTRUCTION
3. GENERALIZED DIAGONAL BAND EXAMPLES
4. SAMPLING PROCEDURE
5. ORDINAL MEASURES OF ASSOCIATION
6. COPULA PARAMETER ELICITATION
7. A VALUE OF INFORMATION EXAMPLE
8. SELECTED REFERENCES

SLIDE 50

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7. ... Description

A VALUE OF INFORMATION EXAMPLE

The farmer needs to

(with a total worth protecting his/her crop of oranges

f $50,000) against freezing weather with the objective of minimizing losses.
Temperature X (in Fahrenheit)

that night. µ YÒ#%ß $%Ó

(below freezing)

Farmer looses entire crop X  $# Ê without protection.

Two protection alternatives:
r

with $

r $

, Burners Sprinklers "!ß !!! $ß !!! respectively, in mobilization cost.

Effectiveness of both is uncertain. Farmer assesses

all-in loss F ÐWÑ to vary between $25000 ($28,000) and $35,000 ($33,000) with a most likely + œ , œ value of $27,000 ($29,000) . 7 œ if it freezes

Assume and

to be F W Ê IÒFÓ œ triangular distributed $29,000ß IÒWÓ œ $30,000.

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7. ... Decision Tree

A VALUE OF INFORMATION EXAMPLE

50

Burners Sprinklers Do Nothing Pr(T < 32)

10 3 E[B] = 29 E[S] = 30

Pr(T > 32) Pr(T < 32) Pr(T > 32) Pr(T < 32) Pr(T > 32)

25.20 24.60 40 24.60

SLIDE 52

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7. ... Dependence on

A VOI EXAMPLE

X

Effectiveness sprinkler

insular layer of freezing

ption is based on an

water on the oranges.

Effectiveness burner

gas usage.

ption is based on
The farmer assesses a

% chance ( % chance) that the burning loss *! '! F (sprinkler loss ) is ( when the temperature W , = Ñ X above it median value

!Þ& !Þ&

is . Hence, we have: below its median value #J T<ÐF  , l X  #Ñ œ !Þ"ß T<ÐW  = l X  #*Ñ œ !Þ%ß

!Þ& !Þ&

where $28,675 and $29,838 , ¸ = ¸ Þ

!Þ& !Þ&

Model dependence between ( ) and using a GDB copula with a power

F W X (slope) generating density with ( ) 8 œ "Î"" œ !Þ% Þ α

SLIDE 53

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7. ... EVPI Freezing

A VOI EXAMPLE

To reduce losses further, the farmer considers consulting either a clairvoyant

Expert A a clairvoyant Expert B

n
r
n

"Freezing" the temperature . X

50

Burners Sprinklers Do nothing

E[B] = 29 E[S] = 30

Pr(T < 32) Burners Sprinklers Do nothing

10 3

Pr(T > 32)

29 23.20 EVPI = 24.6 – 23.2 = 1.4

SLIDE 54

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7. ... EVPI Temperature

A VOI EXAMPLE

X

EVPI on the temperature from Expert B is

since it X more complicated requires evaluation of and IÒFl>Ó IÒWl>ÓÞ

K

> ß IÒFl>Ó = œ #&!! iven we evaluate using realizations using the steps: Step 1: Recall, B œ Ð X µ Y8309<7Ò#%ß $%ÓÑ

>#% $%#%

Step 2: from GDB Sample quantile levels C3ß 3 œ "ß á ß = Ð\ß ] Ñ copula with power generating density for . Ð8Ñ Fß 8 œ "Î"" Step : $ IÒFl>Ó œ L ÐC Ñß

" =3œ" = " 3



F µ

Ð à à Ñ L Ð † Ñ Triang $25,000 $27,000 $35,000 , is the inverse cdf or

"

quantile function of $28,000 $29,000 $33,000 . FÞ W µ X<3+81Ð à à Ñ Evaluation of is analogous. IÒWl>Ó

SLIDE 55

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7. ... EVPI Temperature

A VOI EXAMPLE

X

.

X œ ÐXlX  $#Ñ µ Ò#%ß $#Ó X µ Ò#%ß $%Ó

w

U since U

50

Burners Sprinklers Do nothing

E[B|T’=t] E[S|T’=t]

Pr(T < 32) Burners Sprinklers Do nothing

10 5

Pr(T > 32)

T’ T’=t<32 EVPI = 24.6 – 22.96 = 1.64 28.70 22.96

SLIDE 56

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7. ... EVPI Temperature

A VOI EXAMPLE

X

$25 $27 $29 $31 $33 $35 24.0 26.0 28.0 30.0 32.0

Temperature (in F) Loss (in $1000's) E[B|t] E[S|t]

29 ]} | [ { ] [

'

= = t B E E B E

T

30 ]} | [ { ] [

'

= = t S E E S E

T

S B

7 . 28 ]} ' | [ ], ' | [ ( {

'

= T S E T B E Min ET

SLIDE 57

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7. ... EVPI Temperature

A VOI EXAMPLE

X

EVPI "freezing"

$1, , EVPI "freezing" $1,64 ¸ %!! ¸ !

Summarizing, the farmer is willing to pay $240 dollars more for perfect

information on the temperature . X

Optimal decision switches to Sprinkler option when

provides Expert A "Freezing" information.

Optimal decision switches to Burner option when

provides Expert B "temperature " information, where . > #'  >  $#

When

provides "temperature " information, where the Expert B > #%  >  #'ß

ptimal decision remains the Sprinkler option.

QUESTIONS?

SLIDE 58

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8. ... Selected References

GDB COPULAE

Bojarski, J. . (2001). A new class of band copulas - distributions with uniform marginals Technical Publication, Institute of Mathematics, Technical University of Zielona Góra. Cooke, R.M. and Waij, R. Ð Ñ 1986 Þ Monte carlo sampling for generalized knowledge dependence, with application to human reliability. , 6 (3), pp. 335-343. Risk Analysis Genest, C. and Mackay, J. (1986). The joy of copulas, bivariate distributions with uniform marginals. 40 (4), pp. 280-283. The American Statistician, Ferguson, T.F. . (1995). A class of symmetric bivariate uniform distributions Statistical Papers, 36 (1), pp. 31-40.

S. Kotz and J.R. van Dorp

. (2010) Generalized Diagonal Band Copulas with Two-Sided Generating Densities, , Vol. 7, No. 2, pp. 196-214. Decision Analysis Nelsen, R.B. . Springer, New York. (1999). An Introduction to Copulas Sklar, A. . (1959). Fonctions de répartition à n dimensions et leurs marges Publ. Inst.

Statist. Univ. Paris, 8, pp. 229-231.

Van Dorp, J.R and Kotz, S. (2003). Generalizations of two sided power distributions and their convolution. , 32 (9), pp. 1703 - Communications in Statistics: Theory and Methods 1723.