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

## Description

Maximum likelihood estimation of the 2-parameter inverse Lomax distribution.

## Usage

 ```1 2 3``` ```inv.lomax(lscale = "loglink", lshape2.p = "loglink", iscale = NULL, ishape2.p = NULL, imethod = 1, gscale = exp(-5:5), gshape2.p = exp(-5:5), probs.y = c(0.25, 0.5, 0.75), zero = "shape2.p") ```

## Arguments

 `lscale, lshape2.p` Parameter link functions applied to the (positive) parameters b, and p. See `Links` for more choices. `iscale, ishape2.p, imethod, zero` See `CommonVGAMffArguments` for information. For `imethod = 2` a good initial value for `ishape2.p` is needed to obtain a good estimate for the other parameter. `gscale, gshape2.p` See `CommonVGAMffArguments` for information. `probs.y` See `CommonVGAMffArguments` for information.

## Details

The 2-parameter inverse Lomax distribution is the 4-parameter generalized beta II distribution with shape parameters a=q=1. It is also the 3-parameter Dagum distribution with shape parameter a=1, as well as the beta distribution of the second kind with q=1. More details can be found in Kleiber and Kotz (2003).

The inverse Lomax distribution has density

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

for b > 0, p > 0, y >= 0. Here, b is the scale parameter `scale`, and `p` is a shape parameter. The mean does not seem to exist; the median is 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.

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

## Examples

 ```1 2 3 4 5 6 7``` ```idata <- data.frame(y = rinv.lomax(n = 2000, scale = exp(2), exp(1))) fit <- vglm(y ~ 1, inv.lomax, data = idata, trace = TRUE) fit <- vglm(y ~ 1, inv.lomax(iscale = exp(3)), data = idata, trace = TRUE, epsilon = 1e-8, crit = "coef") coef(fit, matrix = TRUE) Coef(fit) summary(fit) ```

### Example output

```Loading required package: stats4
VGLM    linear loop  1 :  loglikelihood = -10346.0237
VGLM    linear loop  2 :  loglikelihood = -10346.0192
VGLM    linear loop  3 :  loglikelihood = -10346.0192
VGLM    linear loop  1 :  coefficients = 2.79976273, 0.38282768
VGLM    linear loop  2 :  coefficients = 2.333349295, 0.731457743
VGLM    linear loop  3 :  coefficients = 2.174806691, 0.871736357
VGLM    linear loop  4 :  coefficients = 2.147124033, 0.895404231
VGLM    linear loop  5 :  coefficients = 2.145902682, 0.896373509
VGLM    linear loop  6 :  coefficients = 2.145872449, 0.896395146
VGLM    linear loop  7 :  coefficients = 2.145871720, 0.896395663
VGLM    linear loop  8 :  coefficients = 2.145871702, 0.896395676
VGLM    linear loop  9 :  coefficients = 2.145871702, 0.896395676
loge(scale) loge(shape2.p)
(Intercept)    2.145872      0.8963957
scale shape2.p
8.549491 2.450754

Call:
vglm(formula = y ~ 1, family = inv.lomax(iscale = exp(3)), data = idata,
trace = TRUE, epsilon = 1e-08, crit = "coef")

Pearson residuals:
Min      1Q  Median     3Q     Max
loge(scale)     -1.378 -0.9121 -0.0466 0.8319 11.5122
loge(shape2.p) -17.152  0.1785  0.3115 0.3934  0.4253

Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept):1  2.14587    0.10398   20.64   <2e-16 ***
(Intercept):2  0.89640    0.07716   11.62   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Number of linear predictors:  2

Names of linear predictors: loge(scale), loge(shape2.p)

Log-likelihood: -10346.02 on 3998 degrees of freedom

Number of iterations: 9
```

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