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 0.7-9 Index]