of 285

• date post

27-Feb-2018
• Category

## Documents

• view

221

0

Embed Size (px)

### Transcript of Copula Example

• 7/25/2019 Copula Example

1/285

Example of a Constructed Copula Donald F. Behan and Sam Cox Preliminary Draft of Ma

#inear Correlation 0.6

Normal Copula

!.!!! !.!\$ !.!%! !.!! !.%!! !.8!! !.&%!

1.!!! 0.000 0.023 0.050 0.200 0.500 0.800 0.950

!.&"" 0.000 0.023 0.050 0.200 0.499 0.795 0.936

!.&%! 0.000 0.023 0.050 0.200 0.497 0.784 0.916

!.8!! 0.000 0.023 0.050 0.196 0.468 0.699 0.784

!.%!! 0.000 0.022 0.047 0.168 0.352 0.468 0.497

!.!! 0.000 0.017 0.034 0.099 0.168 0.196 0.200

!.!%! 0.000 0.009 0.016 0.034 0.047 0.050 0.050

!.!\$ 0.000 0.005 0.009 0.017 0.022 0.023 0.023

!.!!! 0.000 0.000 0.000 0.000 0.000 0.000 0.000

Normal Copula Density for Common Events - These cells are assumed to be known, and are fixed

!.!!! !.!\$ !.!%! !.!! !.%!! !.8!! !.&%!

1.!!!!.&""

!.&%! 0.004 0.025 0.056 0.047

!.8!! 0.025 0.088 0.116 0.056

!.%!! 0.056 0.116 0.088 0.025

!.!! 0.047 0.056 0.025 0.004

!.!%!

!.!\$

!.!!!

Constructed Copula Densities for Unusual Events

Start by copying cells from right to C34!40 e"cept #35\$38.

%his creates a feasible sol&tion belo'( 'hich can be mo)ifie)( s&b*ect to nonnegati+ity.

,)*&stable cells are in C34-!40 e"cept for #35-\$38( 'hich come from the no'n cop&la for common e+ents.

!.!!! !.!\$ !.!%! !.!! !.%!! !.8!! !.&%!

1.!!! comp&te) comp&te) comp&te) comp&te) comp&te) comp&te)

!.&"" 0.000 0.000 0.000 0.002 0.008 0.010

!.&%! 0.000 0.000

!.8!! 0.001 0.002

!.%!! 0.005 0.008

!.!! 0.008 0.010

!.!%! 0.003 0.003 0.010 0.008 0.002 0.000

!.!\$ 0.005 0.003 0.008 0.005 0.001 0.000

!.!!!

inal Constructed Copula

%hese +al&es are c&m&lati+e probabilities. %hey are base) on the core +al&es in #20-\$23( the )ensities in C34-!40 a

!.!!! !.!\$ !.!%! !.!! !.%!! !.8!! !.&%!

1.!!! 0.000 0.023 0.050 0.200 0.500 0.800 0.950

!.&"" 0.000 0.023 0.050 0.200 0.499 0.795 0.936

!.&%! 0.000 0.023 0.050 0.200 0.497 0.784 0.916

• 7/25/2019 Copula Example

2/285

!.8!! 0.000 0.023 0.050 0.196 0.468 0.699 0.784

!.%!! 0.000 0.022 0.047 0.168 0.352 0.468 0.497

!.!! 0.000 0.017 0.034 0.099 0.168 0.196 0.200

!.!%! 0.000 0.009 0.016 0.034 0.047 0.050 0.050

!.!\$ 0.000 0.005 0.009 0.017 0.022 0.023 0.023

!.!!! 0.000 0.000 0.000 0.000 0.000 0.000 0.000

Calculations for !pearman"s rho

!.!!! !.!\$ !.!%! !.!! !.%!! !.8!! !.&%!

1.!!! 6.693#08 2.562#07 4.795#06 2.545#05 3.951#05 1.752#05

!.&"" 2.562#07 9.796#07 1.821#05 9.455#05 1.408#04 5.893#05

!.&%! 4.795#06 1.821#05 3.229#04 1.516#03 1.945#03 6.777#04

!.8!! 2.545#05 9.455#05 1.516#03 6.158#03 6.692#03 1.945#03

!.%!! 3.951#05 1.408#04 1.945#03 6.692#03 6.158#03 1.516#03

!.!! 1.752#05 5.893#05 6.777#04 1.945#03 1.516#03 3.229#04

!.!%! 1.982#06 6.240#06 5.893#05 1.408#04 9.455#05 1.821#05

!.!\$ 6.447#07 1.982#06 1.752#05 3.951#05 2.545#05 4.795#06!.!!!

1 2 3 4 5 6 7

%he cells belo' are )ensities comp&te) on the basis of the properties of a cop&la.

!f these cells are not nonnegati+e( the a)*&stable cells in C34-!40 ha+e +iolate) the constraints.

Computed Cells

!.!!! !.!\$ !.!%! !.!! !.%!! !.8!! !.&%!

1.!!! 3.011#07 1.659#06 6.267#05 8.811#04 4.528#03 8.335#03

!.&""

!.&%!

!.8!!

!.%!!

!.!!

!.!%!

!.!\$

!.!!!

#pproximate solution to the maximi\$ation of rho when the core values are derived from the norm

!.!!! !.!\$ !.!%! !.!! !.%!! !.8!! !.&%!

1.!!!

!.&"" 0.000 0.000 0.000 0.000 0.016 0.012

!.&%! 0.000 0.000!.8!! 0.000 0.000

!.%!! 0.000 0.016

!.!! 0.007 0.012

!.!%! 0.000 0.000 0.012 0.016 0.000 0.000

!.!\$ 0.016 0.000 0.007 0.000 0.000 0.000

!.!!!

• 7/25/2019 Copula Example

3/285

#pproximate solution to the minimi\$ation of rho when the core values are derived from the norma

!.!!! !.!\$ !.!%! !.!! !.%!! !.8!! !.&%!

1.!!!

!.&"" 0.000 0.000 0.012 0.016 0.000 0.000

!.&%! 0.007 0.012

!.8!! 0.000 0.016

!.%!! 0.000 0.000

!.!! 0.000 0.000

!.!%! 0.000 0.000 0.000 0.000 0.016 0.012

!.!\$ 0.000 0.000 0.000 0.000 0.000 0.007

!.!!!

• 7/25/2019 Copula Example

4/285

18, !!"

!.&"" 1.!!!

0.977 1.000

0.960 0.977

0.936 0.950

0.795 0.800

0.499 0.500

0.200 0.200

0.050 0.050

0.023 0.023

0.000 0.000

Normal Copula Densities

!.&"" 1.!!!

3.144#07 1.664#06 6.268#05 8.811#041.664#06 7.548#06 2.155#04 2.193#03

6.268#05 2.155#04 3.931#03 2.454#02

8.811#04 2.193#03 2.454#02 8.786#02

4.528#03 7.998#03 5.613#02 1.159#01

8.335#03 1.025#02 4.653#02 5.613#02

3.441#03 3.140#03 1.025#02 7.998#03

5.500#03 3.441#03 8.335#03 4.528#03

Test for non-ne(ative

Numerical )alues of Normal Copula Density *startin(

Copy from 19-S25 abo+e &sing paste special( +al&es

!.&"" 1.!!! !.!"%!1\$1& !.!% !. !.%

comp&te) comp&te) 1

0.003 comp&te) !.&""% 1.66442#006 7.548#006 0.0002155 0.0021926328

0.010 comp&te) !.&% 6.26877#005 0.00021551 0.0039307 0.0245405088

0.008 comp&te) !.8 0.0008811259 0.00219261 0.0245405 0.0878568645

0.002 comp&te) !.%0.0045276612 0.00799757 0.0561276 0.1158759477

0.000 comp&te) !.0.0083352143 0.01025374 0.0465341 0.0561276434

0.000 comp&te) !.!%0.0034415009 0.00313973 0.0102537 0.0079975702

0.000 comp&te) !.!"% 0.0054999765 0.0034415 0.0083352 0.0045276739

Density )alues Derived from inal Constructed Copu

n) the comp&te) +al&es in C74-/81.

!.&"" 1.!!! 1 2 3 4

0.977 1.000 3.01094#007 1.659#006 6.267#005 0.0008811062

0.960 0.977 1.66442#006 7.548#006 0.0002155 0.0021926328

0.936 0.950 6.26877#005 0.00021551 0.0039307 0.0245404967

• 7/25/2019 Copula Example

5/285

0.795 0.800 0.0008811259 0.00219261 0.0245405 0.0878568993

0.499 0.500 0.0045276612 0.00799757 0.0561276 0.1158759807

0.200 0.200 0.0083352143 0.01025374 0.046534 0.0561276402

0.050 0.050 0.0034415009 0.00313973 0.0102537 0.0079975702

0.023 0.023 0.0054999765 0.0034415 0.0083352 0.0045276739

0.000 0.000

!.&"" 1.!!! #pproximation to !pearman"s rho

1.982#06 6.447#07 !.%18!1%

6.240#06 1.982#06

5.893#05 1.752#05

1.408#04 3.951#05

9.455#05 2.545#05

1.821#05 4.795#06

9.796#07 2.562#07

2.562#07 6.693#08

8 9

!.&"" 1.!!!

3.441#03 5.500#03

3.441#03

8.335#03

4.528#03

8.811#04

6.267#05

1.649#06

3.098#07

l copula with linear correlation %&'&

!.&"" 1.!!!

0.000

0.0120.016

0.000

0.000

0.000

0.000

• 7/25/2019 Copula Example

6/285

l copula with linear correlation %&'&

!.&"" 1.!!!

0.000

0.000

0.000

0.016

0.012

0.000

0.000

• 7/25/2019 Copula Example

7/285

4.528#03 8.335#03 3.441#03 5.500#037.998#03 1.025#02 3.140#03 3.441#03

5.613#02 4.653#02 1.025#02 8.335#03

1.159#01 5.613#02 7.998#03 4.528#03

8.786#02 2.454#02 2.193#03 8.811#04

2.454#02 3.931#03 2.155#04 6.268#05

2.193#03 2.155#04 7.548#06 1.664#06

8.811#04 6.268#05 1.664#06 3.141#07

point for constructed portion+

!.8 !.&% !.&""'&& 1

0.00799757 0.0102537 0.0031397

0.05612764 0.0465341 0.0102537

0.11587595 0.0561276 0.0079976

0.08785686 0.0245405 0.0021926

0.02454051 0.0039307 0.0002155

0.00219263 0.0002155 7.548#006

0.00088111 6.268#005 1.664#006

la

5 6 7 8

0.00452767 0.0083352 0.0034415 0.0055

0.00799757 0.0102537 0.0031397 0.0034415

0.05612764 0.046534 0.0102537 0.0083352

• 7/25/2019 Copula Example

8/285

0.11587598 0.0561276 0.0079976 0.0045277

0.0878569 0.0245405 0.0021926 0.0008811

0.0245405 0.0039307 0.0002155 6.27#005

0.00219263 0.0002155 7.548#006 1.65#006

0.00088111 6.268#005 1.664#006 3.10#007

• 7/25/2019 Copula Example

9/285

• 7/25/2019 Copula Example

10/285

Simulation of uniform random (aria)les in copula

ase) on cop&la in Constr&cte) Cop&la ,44-/53

random (aria)le u1 0.967677 &niform on 0(1

random (aria)le u 0.522771 &niform on 0(1

lo*er )ound for u1 0.95upper )ound for u1 0.97725 e fin) the &pper bo&n) by looing for the negati+e of &1 in an asc

u1 column 7 %his is the col&mn of the cop&la )etermine) by the ran)om +ariable

u1 column total density 0.02725

scaled u 0.014245 &2 is scale) so that it no' is &niform in 0() 'here ) is t

%he scale) +al&es represent c&m&lati+e probabilities of

%he &nscale) c&m&lati+e probability of ; is fo&n) belo'

s&m ; negati+e

" col. )ens re+erse) c&m prob s&m

+ lo*er pro)a)ility 0.8 1 0.003441 0 0 0.02725

+ upper pro)a)ility 0.95 2 0.00314 1.7#006 0.02275 0.023808

+ lo*er (alue 0