lomax {VGAM} | R Documentation

## Lomax Distribution Family Function

### Description

Maximum likelihood estimation of the 2-parameter Lomax distribution.

### Usage

    lomax(link.scale = "loge", link.q = "loge",
          earg.scale=list(), earg.q=list(),
          init.scale = NULL, init.q = 1, zero = NULL)

### Arguments

`link.scale, link.q` | Parameter link function applied to the (positive) parameters `scale` and `q`. See `Links` for more choices.
`earg.scale, earg.q` | List. Extra argument for each of the links. See `earg` in `Links` for general information.
`init.scale, init.q` | Optional initial values for `scale` and `q`.
`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} which correspond to `scale`, `q`, respectively.

### Details

The 2-parameter Lomax distribution is the 4-parameter generalized beta II distribution with shape parameters a=p=1. It is probably more widely known as the Pareto (II) distribution. It is also the 3-parameter Singh-Maddala distribution with shape parameter a=1, as well as the beta distribution of the second kind with p=1. More details can be found in Kleiber and Kotz (2003).

The Lomax distribution has density

f(y) = q / [b (1 + y/b)^(1+q)]

for b > 0, q > 0, y > 0. Here, b is the scale parameter `scale`, and `q` is a shape parameter. The cumulative distribution function is

F(y) = 1 - [1 + (y/b)]^(-q).

The mean is

E(Y) = b/(q-1)

provided q > 1.

### 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`.

### 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

`Lomax`, `genbetaII`, `betaII`, `dagum`, `sinmad`, `fisk`, `invlomax`, `paralogistic`, `invparalogistic`.

### Examples

    y = rlomax(n=2000, 6, 2)
    fit = vglm(y ~ 1, lomax, trace=TRUE)
    fit = vglm(y ~ 1, lomax, trace=TRUE, crit="c")
    coef(fit, mat=TRUE)
    Coef(fit)
    summary(fit)

[Package VGAM version 0.7-9 Index]