R: Dagum Distribution Family Function

dagum {VGAM} | R Documentation

## Dagum Distribution Family Function

### Description

Maximum likelihood estimation of the 3-parameter Dagum distribution.

### Usage

    dagum(lscale = "loge", lshape1.a = "loge", lshape2.p = "loge",
          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 >= 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`.

### 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.

### Author(s)

T. W. Yee

### 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`, `inv.lomax`, `lomax`, `paralogistic`, `inv.paralogistic`, `simulate.vlm`.

### Examples

    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)

[Package VGAM version 1.0-3 Index]