The 3-parameter Dagum distribution is the 4-parameter
generalized beta II distribution with shape parameter \(q=1\).
It is known under various other names, such as the Burr III,
inverse Burr, beta-K, and 3-parameter kappa distribution.
It can be considered a generalized log-logistic distribution.
Some distributions which are special cases of the 3-parameter
Dagum are the inverse Lomax (\(a=1\)), Fisk (\(p=1\)),
and the inverse paralogistic (\(a=p\)).
More details can be found in Kleiber and Kotz (2003).
The Dagum distribution has a cumulative distribution function
$$F(y) = [1 + (y/b)^{-a}]^{-p}$$
which leads to a probability density function
$$f(y) = ap y^{ap-1} / [b^{ap} \{1 + (y/b)^a\}^{p+1}]$$
for \(a > 0\), \(b > 0\), \(p > 0\), \(y \geq 0\).
Here, \(b\) is the scale parameter scale,
and the others are shape parameters.
The mean is
$$E(Y) = b \, \Gamma(p + 1/a) \, \Gamma(1 - 1/a) / \Gamma(p)$$
provided \(-ap < 1 < a\); these are returned as the fitted
values. This family function handles multiple responses.