Parameter link functions applied to the
(positive) parameters a and scale.
See Links for more choices.
earg.a, earg.scale
List. Extra argument for each of the links.
See earg in Links for general information.
init.a, init.scale
Optional initial values for a and scale.
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} which correspond to
a, scale, respectively.
Value
An object of class "vglmff" (see vglmff-class).
The object is used by modelling functions such as vglm,
and vgam.
Details
The 2-parameter Fisk (aka log-logistic) distribution is the 4-parameter
generalized beta II distribution with shape parameter $q=p=1$.
It is also the 3-parameter Singh-Maddala distribution
with shape parameter $q=1$, as well as the
Dagum distribution with $p=1$.
More details can be found in Kleiber and Kotz (2003).
The Fisk distribution has density
$$f(y) = a y^{a-1} / [b^a {1 + (y/b)^a}^2]$$
for $a > 0$, $b > 0$, $y > 0$.
Here, $b$ is the scale parameter scale,
and a is a shape parameter.
The cumulative distribution function is
$$F(y) = 1 - [1 + (y/b)^a]^{-1} = [1 + (y/b)^{-a}]^{-1}.$$
The mean is
$$E(Y) = b \, \Gamma(1 + 1/a) \, \Gamma(1 - 1/a)$$
provided $a > 1$.
References
Kleiber, C. and Kotz, S. (2003)
Statistical Size Distributions in Economics and
Actuarial Sciences,
Hoboken, NJ: Wiley-Interscience.