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

[Mazensayed91]

Reference issue

What does this implement/fix?

Additional information

[Mazensayed91]

This related to #13996 issue.
I just added another definition for the fisk pdf

[tupui]

Thanks @Mazensayed91. Although per Matt's comment, we should switch the two formulas. Have the one without the minus as the default one.

Also there is a comment about this being also relevant for Burr. If this is correct, we could fix this here as well.

[Mazensayed91]

Thanks @tupui for the modifications, I've modified the request to fit them

[ev-br]

These two formulas are only equivalent for d=1, i.e. for the fisk distribution.

[Mazensayed91]

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

[rgommers]

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.

[mdhaber]

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:

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

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.

[rgommers]

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:

[mdhaber]

In case others would like to check, here are comparisons of how the documentation now renders VS Wikipedia.

Fisk

Wikipedia (log-logistic distribution)

SciPy (fisk)

Burr Type III

Wikipedia (Dagum distribution)

It is not obvious how the CDF would turn into the PDF above or how the PDF shown here would be equivalent to the PDF in SciPy. (Sympy cannot confirm it, so I'm not even sure if the PDF on Wikipedia is correct.) But it is more obvious that the derivative of the CDF above, which is the same as that in Burr's paper, is equivalent to the PDF used in SciPy.

from sympy import symbols, diff, simplify
x, a, p, c, d = symbols('x, a, p, c, d', real=True, positive=True)
F = (1 + x**(-a))**(-p)  # Wikipedia CDF
f = diff(F, x).subs(a, c).subs(p, d)  # Wikipdia deriviative of CDF, SciPy Notation
f_s = c*d*(x**(-c-1))/(1+x**(-c))**(d+1)  # SciPy PDF
print(simplify(f - f_s))  # 0

SciPy

Burr Type XII

Wikipedia ("Burr Distribution")

SciPy (burr12)

I see that it says burr instead of burr12 above; I'll fix that separately rather than waiting for CI to re-run here. This typo was not introduced in this PR.

I'm going to go ahead and merge this, but please let me know if you notice anything that the updates did not catch.

[mdhaber]

Thanks @Mazensayed91 and reviewers!