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

## Description

Maximum likelihood estimation of the 2-parameter Lomax distribution.

## Usage

 ```1 2 3 4``` ```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 >= 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

 ```1 2 3 4 5``` ```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) ```

### Example output

```Loading required package: stats4
VGLM    linear loop  1 :  loglikelihood = -159.2663
VGLM    linear loop  2 :  loglikelihood = -158.01733
VGLM    linear loop  3 :  loglikelihood = -157.96819
VGLM    linear loop  4 :  loglikelihood = -157.96804
VGLM    linear loop  5 :  loglikelihood = -157.96804
loge(scale) loge(shape3.q)
(Intercept)    1.005489       1.984914
scale shape3.q
2.733245 7.278421

Call:
vglm(formula = y ~ 1, family = lomax, data = ldata, trace = TRUE)

Pearson residuals:
Min      1Q   Median     3Q    Max
loge(scale)     -9.668 -0.8674 -0.07777 0.8655 1.5803
loge(shape3.q) -17.451  0.1186  0.24752 0.3377 0.3753

Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept):1   1.0055     0.2956   3.402 0.000669 ***
(Intercept):2   1.9849     0.2618   7.582  3.4e-14 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Number of linear predictors:  2

Names of linear predictors: loge(scale), loge(shape3.q)

Log-likelihood: -157.968 on 1998 degrees of freedom

Number of iterations: 5
```

VGAM documentation built on Jan. 16, 2021, 5:21 p.m.