<abstract xmlns="http://www.w3.org/1999/xhtml">

Two-dimensional distribution function <italic>g</italic>(<italic>x, y</italic>) defined in [0, 1]2 is called copula, if <italic>g</italic>(<italic>x,</italic> 1) = <italic>x</italic> and <italic>g</italic>(1,<italic>y</italic>)= <italic>y</italic> for every <italic>x, y</italic>. Similarly, <italic>s</italic>-dimensional copula is a distribution function <italic>g</italic>(<italic>x</italic>1,<italic>x</italic>2,...,<italic>x</italic>s) such that every <italic>k</italic>-dimensional face function
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2023-0009_eq_001.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mi>g</mi><mrow><mo>(</mo><mrow><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mn>1</mn><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>i</mi></mrow><mn>1</mn></msub></mrow></msub><mo>,</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mn>1</mn><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>i</mi></mrow><mn>2</mn></msub></mrow></msub><mo>,</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mn>1</mn><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>i</mi></mrow><mi>k</mi></msub></mrow></msub><mo>,</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></math>
<tex-math>g\left( {1, \ldots ,1,{x_{{i_1}}},1, \ldots ,1,{x_{{i_2}}},1, \ldots ,1,{x_{{i_k}}},1, \ldots ,1} \right)</tex-math>
</alternatives>
</disp-formula>
is equal to <italic>x</italic>i1 <italic>x</italic>i2 ...<italic>x</italic>ik for some but fixed <italic>k</italic>. In this paper we summarize and extend all known parts of copulas.
In this paper we use the following abbreviations:
<list list-type="simple">
<list-item>{x} — <italic>fractional part of x</italic>;</list-item>
<list-item>{x} — <italic>x mod</italic> 1;</list-item>
<list-item>[x] — <italic>integer part of x</italic>;</list-item>
<list-item>u.d. — <italic>uniform distribution</italic>;</list-item>
<list-item>d.f. — <italic>distribution function</italic>;</list-item>
<list-item>a.d.f. — <italic>asymptotic distribution function</italic>;</list-item>
<list-item>u.d.p. — <italic>uniform distribution preserving</italic>;</list-item>
<list-item>step d.f. — <italic>step distribution function</italic>;</list-item>
<list-item>a.e. — <italic>almost everywhere</italic>;</list-item>
<list-item>#X — <italic>cardinality of the set X</italic>.</list-item>
</list>
</abstract>

Two-dimensional distribution function g(x, y) defined in [0, 1]2 is called copula, if g(x, 1) = x and g(1,y)= y for every x, y. Similarly, s-dimensional copula is a distribution function g(x1,x2,...,xs) such that every k-dimensional face function



g(1,…,1,xi1,1,…,1,xi2,1,…,1,xik,1,…,1)
g\left( {1, \ldots ,1,{x_{{i_1}}},1, \ldots ,1,{x_{{i_2}}},1, \ldots ,1,{x_{{i_k}}},1, \ldots ,1} \right)


is equal to xi1 xi2 ...xik for some but fixed k. In this paper we summarize and extend all known parts of copulas.
In this paper we use the following abbreviations:

{x} — fractional part of x;
{x} — x mod 1;
[x] — integer part of x;
u.d. — uniform distribution;
d.f. — distribution function;
a.d.f. — asymptotic distribution function;
u.d.p. — uniform distribution preserving;
step d.f. — step distribution function;
a.e. — almost everywhere;
#X — cardinality of the set X.

Copulas

Slovak University of Technology in Bratislava

Uniform distribution theory

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

{"article-title":"Copulas"}

Two-dimensional distribution function g(x, y) defined in [0, 1]2 is called copula, if g(x, 1) = x and g(1,y)= y for every x, y. Similarly, s-dimensional copula...

Copulas

Published Online: Aug 10, 2023

Page range: 147 - 200

Received: Mar 24, 2023

Accepted: Jun 22, 2023

DOI: https://doi.org/10.2478/udt-2023-0009

Keywords
distribution function, fractional part, copula

© 2023 Oto Strauch et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Copulas

Published Online: Aug 10, 2023

Page range: 147 - 200

Received: Mar 24, 2023

Accepted: Jun 22, 2023

DOI: https://doi.org/10.2478/udt-2023-0009

Keywordsdistribution function, fractional part, copula

© 2023 Oto Strauch et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Keywords
distribution function, fractional part, copula