Dagum Distribution Family Function

Maximum likelihood estimation of the 3-parameter Dagum distribution.

dagum(lscale = "loglink", lshape1.a = "loglink", lshape2.p =
      "loglink", iscale = NULL, ishape1.a = NULL, ishape2.p =
      NULL, imethod = 1, lss = TRUE, gscale = exp(-5:5), gshape1.a
      = seq(0.75, 4, by = 0.25), gshape2.p = exp(-5:5), probs.y =
      c(0.25, 0.5, 0.75), zero = "shape")

Arguments

lss

See CommonVGAMffArguments for important information.

lshape1.a, lscale, lshape2.p

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

iscale, ishape1.a, ishape2.p, imethod, zero

See CommonVGAMffArguments for information. For imethod = 2 a good initial value for ishape2.p is needed to obtain a good estimate for the other parameter.

gscale, gshape1.a, gshape2.p

See CommonVGAMffArguments for information.

probs.y

See CommonVGAMffArguments for information.

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. This family function handles multiple responses.

Value

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

References

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

Author

T. W. Yee

Note

See the notes in genbetaII.

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.

Examples

if (FALSE) { # \dontrun{
ddata <- data.frame(y = rdagum(n = 3000, scale = exp(2),
                               shape1 = exp(1), shape2 = exp(1)))
fit <- vglm(y ~ 1, dagum(lss = FALSE), data = ddata, trace = TRUE)
fit <- vglm(y ~ 1, dagum(lss = FALSE, ishape1.a = exp(1)),
            data = ddata, trace = TRUE)
coef(fit, matrix = TRUE)
Coef(fit)
summary(fit)
} # }