VGAM (version 1.0-4)

# lomax: 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`.

## Value

An object of class `"vglmff"` (see `vglmff-class`). The object is used by modelling functions such as `vglm`, and `vgam`.

## 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.

## 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

Run this code
```# NOT RUN {
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)
# }
```

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