VGAM (version 0.9-1)

dagum: Dagum Distribution Family Function

Description

Maximum likelihood estimation of the 3-parameter Dagum distribution.

Usage

dagum(lshape1.a = "loge", lscale = "loge", lshape2.p = "loge",
      ishape1.a = NULL, iscale = NULL, ishape2.p = 1, zero = NULL)

Arguments

lshape1.a, lscale, lshape2.p

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

ishape1.a, iscale, ishape2.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 $a > 0$, $b > 0$, $p > 0$, $y \geq 0$. 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 $-ap < 1 < a$; these are returned as the fitted values.

References

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

See Also

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

Examples

ddata <- data.frame(y = rdagum(n = 3000, exp(1), exp(2), exp(1)))
fit <- vglm(y ~ 1, dagum, ddata, trace = TRUE)
fit <- vglm(y ~ 1, dagum(ishape1.a = exp(1)), ddata, trace = TRUE)
coef(fit, matrix = TRUE)
Coef(fit)
summary(fit)