lomax {VGAM} | R Documentation

## Lomax Distribution Family Function

### Description

Maximum likelihood estimation of the 2-parameter Lomax distribution.

### Usage

    lomax(lscale = "loge", lshape3.q = "loge", iscale = NULL,
          ishape3.q = NULL, imethod = 1, gscale = exp(-5:5),
          gshape3.q = seq(0.75, 4, by = 0.25),
          probs.y = c(0.25, 0.5, 0.75), zero = "shape")

### Arguments

`lscale, lshape3.q` | Parameter link function applied to the (positive) parameters `scale` and `q`. See `Links` for more choices.
`iscale, ishape3.q, imethod` | See `CommonVGAMffArguments` for information. For `imethod = 2` a good initial value for `iscale` is needed to obtain a good estimate for the other parameter.
`gscale, gshape3.q, zero, probs.y` | See `CommonVGAMffArguments`.

### 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; 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`.

### 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

`Lomax`, `genbetaII`, `betaII`, `dagum`, `sinmad`, `fisk`, `inv.lomax`, `paralogistic`, `inv.paralogistic`, `simulate.vlm`.

### Examples

    ldata <- data.frame(y = rlomax(n = 1000, scale =  exp(1), exp(2)))
    fit <- vglm(y ~ 1, lomax, data = ldata, trace = TRUE)
    coef(fit, matrix = TRUE)
    Coef(fit)
    summary(fit)

[Package VGAM version 1.0-3 Index]