1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
|
.. _continuous-dgamma:
Double Gamma Distribution
=========================
The double gamma is the signed version of the Gamma distribution. For :math:`\alpha>0:`
.. math::
:nowrap:
\begin{eqnarray*} f\left(x;\alpha\right) & = & \frac{1}{2\Gamma\left(\alpha\right)}\left|x\right|^{\alpha-1}e^{-\left|x\right|}\\
F\left(x;\alpha\right) & = & \left\{
\begin{array}{ccc}
\frac{1}{2}-\frac{\gamma\left(\alpha,\left|x\right|\right)}{2\Gamma\left(\alpha\right)} & & x\leq0\\
\frac{1}{2}+\frac{\gamma\left(\alpha,\left|x\right|\right)}{2\Gamma\left(\alpha\right)} & & x>0
\end{array}
\right.\\
G\left(q;\alpha\right) & = & \left\{
\begin{array}{ccc}
-\gamma^{-1}\left(\alpha,\left|2q-1\right|\Gamma\left(\alpha\right)\right) & & q\leq\frac{1}{2}\\
\gamma^{-1}\left(\alpha,\left|2q-1\right|\Gamma\left(\alpha\right)\right) & & q>\frac{1}{2}
\end{array}
\right.\end{eqnarray*}
.. math::
M\left(t\right)=\frac{1}{2\left(1-t\right)^{a}}+\frac{1}{2\left(1+t\right)^{a}}
.. math::
:nowrap:
\begin{eqnarray*} \mu=m_{n} & = & 0\\
\mu_{2} & = & \alpha\left(\alpha+1\right)\\
\gamma_{1} & = & 0\\
\gamma_{2} & = & \frac{\left(\alpha+2\right)\left(\alpha+3\right)}{\alpha\left(\alpha+1\right)}-3\\
m_{d} & = & \mathrm{NA}\end{eqnarray*}
Implementation: `scipy.stats.dgamma`
|
