Copula Example

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    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

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    !.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

    !.!!!

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    #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

    !.!!!

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    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

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    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

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    l copula with linear correlation %&'&

    !.&"" 1.!!!

    0.000

    0.000

    0.000

    0.016

    0.012

    0.000

    0.000

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    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

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    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

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    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