dagum {VGAM}R Documentation

Dagum Distribution Family Function

Description

Maximum likelihood estimation of the 3-parameter Dagum distribution.

Usage

dagum(link.a = "loge", link.scale = "loge", link.p = "loge",
      earg.a=list(), earg.scale=list(), earg.p=list(),
      init.a = NULL, init.scale = NULL, init.p = 1, zero = NULL)

Arguments

link.a, link.scale, link.p 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.

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.

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.

From Kleiber and Kotz (2003), the MLE is rather sensitive to isolated observations located sufficiently far from the majority of the data. Reliable estimation of the scale parameter require n>7000, while estimates for a and p can be considered unbiased for n>2000 or 3000.

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

Dagum, genbetaII, betaII, sinmad, fisk, invlomax, lomax, paralogistic, invparalogistic.

Examples

y = rdagum(n=3000, 4, 6, 2)
fit = vglm(y ~ 1, dagum, trace=TRUE)
fit = vglm(y ~ 1, dagum(init.a=2.1), trace=TRUE, crit="c")
coef(fit, mat=TRUE)
Coef(fit)
summary(fit)

[Package VGAM version 0.7-7 Index]