1%feature("docstring") OT::PlackettCopula
2"Plackett copula.
3
4Parameters
5----------
6theta : float
7    Parameter :math:`\theta`, :math:`\theta \geq 0`. Default is :math:`\theta=2`.
8
9Notes
10-----
11The Plackett copula is a bivariate symmetric copula defined by:
12
13.. math::
14
15    C(u_1, u_2) = \frac{\left[1+(\theta-1)(u_1+u_2)\right]-
16                  \sqrt{\left[1+(\theta-1)(u_1+u_2)\right]^2-
17                  4u_1u_2\theta(\theta-1)}}{2(\theta-1)}
18
19for :math:`(u_1, u_2) \in [0, 1]^2`
20
21This copula is the only copula with constant *odd ratio* :math:`\theta\geq 0`:
22
23.. math::
24
25   \theta = \frac{\Prob{U_1\leq u_1,U_2\leq u_2}\Prob{U_1>u_1,U_2>u_2}}{\Prob{U_1\leq u_1,U_2>u_2}\Prob{U_1>u_1,U_2\leq u_2}} = \frac{C(u_1,u_2)\left[1-u_1-u_2+C(u_1,u_2)\right]}{\left[u_1-C(u_1,u_2)\right]\left[u_2-C(u_1,u_2)\right]}
26
27This is a *comprehensive family* of copulas as it contains:
28
29- the Frechet lower bound :math:`W(u_1,u_2)=\max(0,u_1+u_2-1)=\lim_{\theta\rightarrow\infty}C(u_1,u_2)`,
30- the Frechet upper bound :math:`M(u_1,u_2)=\min(u_1,u_2)=C_0(u_1,u_2)`
31- the independent copula :math:`\Pi(u_1,u_2)=u_1u_2=C_1(u_1,u_2)`.
32
33Examples
34--------
35Create a distribution:
36
37>>> import openturns as ot
38>>> copula = ot.PlackettCopula(2.5)
39
40Draw a sample:
41
42>>> sample = copula.getSample(5)"
43
44// ---------------------------------------------------------------------
45
46%feature("docstring") OT::PlackettCopula::getTheta
47"Get the parameter :math:`\theta`.
48
49Returns
50-------
51theta : float
52    Parameter :math:`\theta` of the copula."
53
54// ---------------------------------------------------------------------
55
56%feature("docstring") OT::PlackettCopula::setTheta
57"Set the parameter :math:`\theta`.
58
59Parameters
60----------
61theta : float
62    Parameter :math:`\theta\geq 0` of the copula."
63