Parameter link functions applied to the
(positive) parameters a, scale, and p.
See Links for more choices.
earg.a, earg.scale, earg.p
List. Extra argument for each of the links.
See earg in Links for general information.
init.a, init.scale, init.p
Optional initial values for a, scale, and p.
zero
An integer-valued vector specifying which
linear/additive predictors are modelled as intercepts only.
Here, the values must be from the set {1,2,3} which correspond to
a, scale, p, respectively.
Value
An object of class "vglmff" (see vglmff-class).
The object is used by modelling functions such as vglm,
and vgam.
Details
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 $a > 0$, $b > 0$, $p > 0$, $y > 0$.
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 $-ap < 1 < a$.
References
Kleiber, C. and Kotz, S. (2003)
Statistical Size Distributions in Economics and
Actuarial Sciences,
Hoboken, NJ: Wiley-Interscience.