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I saw on wikipedia https://en.wikipedia.org/wiki/Generalized_logistic_distribution that when $\alpha<\beta$, generalized type IV logistic distribution can be written as:

$\frac{\exp(-\alpha x)}{(\exp(-x)+1)^\beta}$

But I'm wondering if the case $\alpha\ge\beta$ is studied?

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  • $\begingroup$ Could you point out where in that Wikipedia article it requires $\alpha\lt\beta$ for the Type IV distribution? I cannot see that anywhere. $\endgroup$ – whuber Apr 12 at 20:20
  • $\begingroup$ If you reparametrize type IV into numerator power and denominator power you'll get this. @whuber $\endgroup$ – ZHU Apr 12 at 22:06
  • $\begingroup$ Now I see the difference. But notice that this is supposed to be a probability density supported on $x\in\mathbb R:$ what is the integral of this pdf when $\beta \le \alpha$?? $\endgroup$ – whuber Apr 13 at 13:29

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