# lomax: Lomax Distribution Family Function In VGAM: Vector Generalized Linear and Additive Models

 lomax R Documentation

## Lomax Distribution Family Function

### Description

Maximum likelihood estimation of the 2-parameter Lomax distribution.

### Usage

``````lomax(lscale = "loglink", lshape3.q = "loglink", 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 \geq 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`.

T. W. Yee

### References

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

`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)
``````

VGAM documentation built on Sept. 19, 2023, 9:06 a.m.