fisk {VGAM}R Documentation

Fisk Distribution family function

Description

Maximum likelihood estimation of the 2-parameter Fisk distribution.

Usage

fisk(link.a = "loge", link.scale = "loge",
     earg.a=list(), earg.scale=list(),
     init.a = NULL, init.scale = NULL, zero = NULL)

Arguments

link.a, link.scale 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.

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.

Value

An object of class "vglmff" (see vglmff-class). The object is used by modelling functions such as vglm, and vgam.

Note

If the self-starting initial values fail, try experimenting with the initial value arguments, especially those whose default value is not NULL.

Author(s)

T. W. Yee

References

Kleiber, C. and Kotz, S. (2003) Statistical Size Distributions in Economics and Actuarial Sciences, Hoboken, NJ: Wiley-Interscience.

See Also

Fisk, genbetaII, betaII, dagum, sinmad, invlomax, lomax, paralogistic, invparalogistic.

Examples

y = rfisk(n=200, 4, 6)
fit = vglm(y ~ 1, fisk, trace=TRUE)
fit = vglm(y ~ 1, fisk(init.a=3.3), trace=TRUE, crit="c")
coef(fit, mat=TRUE)
Coef(fit)
summary(fit)

[Package VGAM version 0.7-7 Index]