Maximum likelihood estimation of the 4-parameter generalized beta II distribution.

```
genbetaII(lscale = "loge", lshape1.a = "loge", lshape2.p = "loge",
lshape3.q = "loge", iscale = NULL, ishape1.a = NULL,
ishape2.p = NULL, ishape3.q = NULL, lss = TRUE,
gscale = exp(-5:5), gshape1.a = exp(-5:5),
gshape2.p = exp(-5:5), gshape3.q = exp(-5:5),
zero = "shape")
```

lss

See `CommonVGAMffArguments`

for important information.

lshape1.a, lscale, lshape2.p, lshape3.q

Parameter link functions applied to the
shape parameter `a`

,
scale parameter `scale`

,
shape parameter `p`

, and
shape parameter `q`

.
All four parameters are positive.
See `Links`

for more choices.

iscale, ishape1.a, ishape2.p, ishape3.q

Optional initial values for the parameters.
A `NULL`

means a value is computed internally using
the arguments `gscale`

, `gshape1.a`

, etc.

gscale, gshape1.a, gshape2.p, gshape3.q

See `CommonVGAMffArguments`

for information.
Replaced by `iscale`

, `ishape1.a`

etc. if given.

zero

The default is to set all the shape parameters to be
intercept-only.
See `CommonVGAMffArguments`

for information.

An object of class `"vglmff"`

(see `vglmff-class`

).
The object is used by modelling functions such as `vglm`

,
and `vgam`

.

This distribution is very flexible and it is not generally
recommended to use this family function when the sample size is
small---numerical problems easily occur with small samples.
Probably several hundred observations at least are needed in order
to estimate the parameters with any level of confidence.
Neither is the inclusion of covariates recommended at all---not
unless there are several thousand observations.
The mean is finite only when \(-ap < 1 < aq\), and this can be
easily violated by the parameter estimates for small sample sizes.
Try fitting some of the special cases of this distribution
(e.g., `sinmad`

, `fisk`

, etc.) first, and
then possibly use those models for initial values for this
distribution.

This distribution is most useful for unifying a substantial number of size distributions. For example, the Singh-Maddala, Dagum, Fisk (log-logistic), Lomax (Pareto type II), inverse Lomax, beta distribution of the second kind distributions are all special cases. Full details can be found in Kleiber and Kotz (2003), and Brazauskas (2002). The argument names given here are used by other families that are special cases of this family. Fisher scoring is used here and for the special cases too.

The 4-parameter generalized beta II distribution has density
$$f(y) = a y^{ap-1} / [b^{ap} B(p,q) \{1 + (y/b)^a\}^{p+q}]$$
for \(a > 0\), \(b > 0\), \(p > 0\), \(q > 0\), \(y \geq 0\).
Here \(B\) is the beta function, and
\(b\) is the scale parameter `scale`

,
while the others are shape parameters.
The mean is
$$E(Y) = b \, \Gamma(p + 1/a) \, \Gamma(q - 1/a) / (\Gamma(p) \, \Gamma(q))$$
provided \(-ap < 1 < aq\); these are returned as the fitted values.

This family function handles multiple responses.

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

Brazauskas, V. (2002)
Fisher information matrix for the Feller-Pareto distribution.
*Statistics & Probability Letters*,
**59**, 159--167.

`dgenbetaII`

,
`betaff`

,
`betaII`

,
`dagum`

,
`sinmad`

,
`fisk`

,
`lomax`

,
`inv.lomax`

,
`paralogistic`

,
`inv.paralogistic`

,
`lino`

,
`CommonVGAMffArguments`

,
`vglm.control`

.

```
# NOT RUN {
gdata <- data.frame(y = rsinmad(3000, shape1 = exp(1), scale = exp(2),
shape3 = exp(1))) # A special case!
fit <- vglm(y ~ 1, genbetaII(lss = FALSE), data = gdata, trace = TRUE)
fit <- vglm(y ~ 1, data = gdata, trace = TRUE,
genbetaII(ishape1.a = 3, iscale = 7, ishape3.q = 2.3))
coef(fit, matrix = TRUE)
Coef(fit)
summary(fit)
# }
```

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