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I have been searching for dagum type 2 probability distribution function for several hours but all I have found is the cumulative density function of the mentioned distribution which is as follows:

$$F(x) = \delta + (1-\delta)\left(1+\left(\frac{x}{b}\right)^{-a}\right)^{-p}$$

Reference : A Guide to the Dagum Distributions, page 4

Are there any PDF for dagum type 2 and 3 distribution just like type 1?

Sepideh Abadpour
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    Take the derivative. Equivalently, change the variables for the PDF of $\log(x)$ given under "Basic Properties." – whuber Dec 26 '23 at 20:38
  • @whuber so the PDF would be: $$f(x) = \frac{ap(1+\delta)}{b^{-a}}x^{-(a+1)}\left(1+\left(\frac{x}{b}\right)^{-a}\right)^{-(p+1)}$$ – Sepideh Abadpour Dec 27 '23 at 16:36
  • Close. See equation (4) for confirmation of the derivative, but then read the rest of the page to learn why when $\delta \gt 0,$ this distribution does not have a density at $x=0.$ – whuber Dec 27 '23 at 16:42
  • You might find https://stats.stackexchange.com/questions/214485, https://stats.stackexchange.com/questions/335547, and https://stats.stackexchange.com/questions/306491 helpful. – whuber Dec 29 '23 at 15:32
  • @whuber but equation (4) is for type 1 dagum distribution – Sepideh Abadpour Dec 29 '23 at 16:13
  • That contradicts what you write in your question. And I was referring to (4) only to check your differentiation: the text after that equation discusses the type II distribution. Start reading in the middle of the page. – whuber Dec 29 '23 at 17:04

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