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DOC: clarify the definition of the pdf of stats.fisk #14003

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Feb 26, 2022
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DOC: clarify the definition of the pdf of stats.fisk #14003

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41be299
added another definition for the pdf for 'fisk'
Mazensayed91 May 8, 2021
5706376
Update _continuous_distns.py
Mazensayed91 May 8, 2021
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Merge remote-tracking branch 'upstream/main'
mdhaber Feb 23, 2022
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DOC: stats: clarify burr/fisk distribution PDFs
mdhaber Feb 23, 2022
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13 changes: 13 additions & 0 deletions scipy/stats/_continuous_distns.py
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Expand Up @@ -883,6 +883,12 @@ class burr_gen(rv_continuous):
-----
The probability density function for `burr` is:

.. math::

f(x, c, d) = c d x^{c - 1} / (1 + x^{c})^{d + 1}

Also equivalent to:

.. math::

f(x, c, d) = c d x^{-c - 1} / (1 + x^{-c})^{d + 1}
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@ev-br ev-br May 8, 2021 •
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These two formulas are only equivalent for d=1, i.e. for the fisk distribution.

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This is the pdf formula for burr distribution, so the first one is the right one ?

image

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I agree, looks like the first one is the right one. Also, if we're using LaTeX anyway, let's use \frac and make it look identical to how the pdf is given on Wikipedia.

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@mdhaber mdhaber Feb 23, 2022 •
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It looks like the second one, the original one, is the intended one for the burr distribution.

This is the PDF corresponding to the third CDF given in Burr's list;
specifically, it is equation (11) in Burr's paper [1]_

This is:
image

Taking the derivative and changing the symbols used for the parameters, it turns into:
image

This is what the code implements when x != 0:

        output = _lazywhere(x == 0, [x, c, d],
                   lambda x_, c_, d_: c_ * d_ * (x_**(c_*d_-1)) / (1 + x_**c_),
                   f2 = lambda x_, c_, d_: (c_ * d_ * (x_ ** (-c_ - 1.0)) /
                                            ((1 + x_ ** (-c_)) ** (d_ + 1.0))))

(I'll leave simplifying that to another time...)

It seems that the other one is burr12.

Expand Down Expand Up @@ -1079,10 +1085,17 @@ class fisk_gen(burr_gen):
-----
The probability density function for `fisk` is:

.. math::

f(x, c) = c x^{c-1} (1 + x^{c})^{-2}

Also equivalent to:
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This is actually confusing, without reading the linked issue it was non-obvious that this wasn't a mistake. I suggest the following rephrase: Please note that the above expression can be transformed into the following one, which is also commonly used:


.. math::

f(x, c) = c x^{-c-1} (1 + x^{-c})^{-2}


for :math:`x >= 0` and :math:`c > 0`.

`fisk` takes ``c`` as a shape parameter for :math:`c`.
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