# dagum: Dagum Distribution Family Function In VGAM: Vector Generalized Linear and Additive Models

## Description

Maximum likelihood estimation of the 3-parameter Dagum distribution.

## Usage

 ```1 2 3 4``` ```dagum(lscale = "loglink", lshape1.a = "loglink", lshape2.p = "loglink", iscale = NULL, ishape1.a = NULL, ishape2.p = NULL, imethod = 1, lss = TRUE, gscale = exp(-5:5), gshape1.a = seq(0.75, 4, by = 0.25), gshape2.p = exp(-5:5), probs.y = c(0.25, 0.5, 0.75), zero = "shape") ```

## Arguments

 `lss` See `CommonVGAMffArguments` for important information. `lshape1.a, lscale, lshape2.p` Parameter link functions applied to the (positive) parameters `a`, `scale`, and `p`. See `Links` for more choices. `iscale, ishape1.a, 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, gshape1.a, gshape2.p` See `CommonVGAMffArguments` for information. `probs.y` See `CommonVGAMffArguments` for information.

## Details

The 3-parameter Dagum distribution is the 4-parameter generalized beta II distribution with shape parameter q=1. It is known under various other names, such as the Burr III, inverse Burr, beta-K, and 3-parameter kappa distribution. It can be considered a generalized log-logistic distribution. Some distributions which are special cases of the 3-parameter Dagum are the inverse Lomax (a=1), Fisk (p=1), and the inverse paralogistic (a=p). More details can be found in Kleiber and Kotz (2003).

The Dagum distribution has a cumulative distribution function

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

which leads to a probability density function

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

for a > 0, b > 0, p > 0, y >= 0. Here, b is the scale parameter `scale`, and the others are shape parameters. The mean is

E(Y) = b gamma(p + 1/a) gamma(1 - 1/a) / gamma(p)

provided -ap < 1 < a; 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`.

From Kleiber and Kotz (2003), the MLE is rather sensitive to isolated observations located sufficiently far from the majority of the data. Reliable estimation of the scale parameter require n>7000, while estimates for a and p can be considered unbiased for n>2000 or 3000.

T. W. Yee

## References

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

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

## Examples

 ```1 2 3 4 5 6 7 8``` ```ddata <- data.frame(y = rdagum(n = 3000, scale = exp(2), shape1 = exp(1), shape2 = exp(1))) fit <- vglm(y ~ 1, dagum(lss = FALSE), data = ddata, trace = TRUE) fit <- vglm(y ~ 1, dagum(lss = FALSE, ishape1.a = exp(1)), data = ddata, trace = TRUE) coef(fit, matrix = TRUE) Coef(fit) summary(fit) ```

### Example output

```Loading required package: stats4
VGLM    linear loop  1 :  loglikelihood = -9775.41159
VGLM    linear loop  2 :  loglikelihood = -9775.41096
VGLM    linear loop  3 :  loglikelihood = -9775.41096
VGLM    linear loop  1 :  loglikelihood = -9775.41103
VGLM    linear loop  2 :  loglikelihood = -9775.41096
VGLM    linear loop  3 :  loglikelihood = -9775.41096
loge(shape1.a) loge(scale) loge(shape2.p)
(Intercept)      0.9924223     1.97818       1.035124
shape1.a    scale shape2.p
2.697761 7.229571 2.815454

Call:
vglm(formula = y ~ 1, family = dagum(lss = FALSE, ishape1.a = exp(1)),
data = ddata, trace = TRUE)

Pearson residuals:
Min      1Q  Median     3Q    Max
loge(shape1.a)  -6.471 -0.3715 0.37954 0.7236 0.8188
loge(scale)     -2.148 -0.6992 0.34606 0.8717 5.7279
loge(shape2.p) -23.820 -0.2013 0.01137 0.3145 4.7115

Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept):1  0.99242    0.02457  40.390   <2e-16 ***
(Intercept):2  1.97818    0.06113  32.359   <2e-16 ***
(Intercept):3  1.03512    0.11680   8.863   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Number of linear predictors:  3

Names of linear predictors: loge(shape1.a), loge(scale), loge(shape2.p)

Log-likelihood: -9775.411 on 8997 degrees of freedom

Number of iterations: 3
```

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