VGAM (version 0.9-1)

fisk: Fisk Distribution family function

Description

Maximum likelihood estimation of the 2-parameter Fisk distribution.

Usage

fisk(lshape1.a = "loge", lscale = "loge",
     ishape1.a = NULL, iscale = NULL, zero = NULL)

Arguments

lshape1.a, lscale

Parameter link functions applied to the (positive) parameters a and scale. See Links for more choices.

ishape1.a, iscale

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 $a > 0$, $b > 0$, $y \geq 0$. 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 $a > 1$; these are returned as the fitted values.

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

fdata <- data.frame(y = rfisk(n = 200, exp(1), exp(2)))
fit <- vglm(y ~ 1, fisk, fdata, trace = TRUE)
fit <- vglm(y ~ 1, fisk(ishape1.a = exp(1)), fdata, trace = TRUE)
coef(fit, matrix = TRUE)
Coef(fit)
summary(fit)