scipy.stats._continuous_distns:kappa4_gen

Notes

The probability density function for kappa4 is:

if $h$ and $k$ are not equal to 0.

If $h$ or $k$ are zero then the pdf can be simplified:

$h=0$ and $k\ne 0$:

kappa4.pdf(x, h, k) = (1.0 - k*x)**(1.0/k - 1.0)*
                      exp(-(1.0 - k*x)**(1.0/k))

$h\ne 0$ and $k=0$:

kappa4.pdf(x, h, k) = exp(-x)*(1.0 - h*exp(-x))**(1.0/h - 1.0)

$h=0$ and $k=0$:

kappa4.pdf(x, h, k) = exp(-x)*exp(-exp(-x))

kappa4 takes $h$ and $k$ as shape parameters.

The kappa4 distribution returns other distributions when certain $h$ and $k$ values are used.

+------+-------------+----------------+------------------+
| h    | k=0.0       | k=1.0          | -inf<=k<=inf     |
+======+=============+================+==================+
| -1.0 | Logistic    |                | Generalized      |
|      |             |                | Logistic(1)      |
|      |             |                |                  |
|      | logistic(x) |                |                  |
+------+-------------+----------------+------------------+
|  0.0 | Gumbel      | Reverse        | Generalized      |
|      |             | Exponential(2) | Extreme Value    |
|      |             |                |                  |
|      | gumbel_r(x) |                | genextreme(x, k) |
+------+-------------+----------------+------------------+
|  1.0 | Exponential | Uniform        | Generalized      |
|      |             |                | Pareto           |
|      |             |                |                  |
|      | expon(x)    | uniform(x)     | genpareto(x, -k) |
+------+-------------+----------------+------------------+

• There are at least five generalized logistic distributions. Four are described here: https://en.wikipedia.org/wiki/Generalized_logistic_distribution The "fifth" one is the one kappa4 should match which currently isn't implemented in scipy: https://en.wikipedia.org/wiki/Talk:Generalized_logistic_distribution https://www.mathwave.com/help/easyfit/html/analyses/distributions/gen_logistic.html

• This distribution is currently not in scipy.