genbetaII {VGAM} | R Documentation

## Generalized Beta Distribution of the Second Kind

### Description

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

### Usage

    genbetaII(link.a = "loge", link.scale = "loge",
              link.p = "loge", link.q = "loge",
              earg.a=list(), earg.scale=list(), earg.p=list(), earg.q=list(),
              init.a = NULL, init.scale = NULL, init.p = 1, init.q = 1,
              zero = NULL)

### Arguments

`link.a, link.scale, link.p, link.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.
`earg.a, earg.scale, earg.p, earg.q` | List. Extra argument for each of the links. See `earg` in `Links` for general information.
`init.a, init.scale` | Optional initial values for `a` and `scale`. A `NULL` means a value is computed internally.
`init.p, init.q` | Optional initial values for `p` and `q`.
`zero` | An integer-valued vector specifying which linear/additive predictors are modelled as intercepts only. Here, the values must be from the set {1,2,3,4} which correspond to `a`, `scale`, `p`, `q`, respectively.

### Details

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 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 > 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.

### Value

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

### Note

If the self-starting initial values fail, try experimenting with the initial value arguments, especially those whose default value is not `NULL`.

Successful convergence depends on having very good initial values. This is rather difficult for this distribution! More improvements could be made here.

### Author(s)

T. W. Yee

### References

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

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

### See Also

`lino`, `betaff`, `betaII`, `dagum`, `sinmad`, `fisk`, `lomax`, `invlomax`, `paralogistic`, `invparalogistic`.

### Examples

    y = rsinmad(n=3000, 4, 6, 2) # Not very good data!
    fit = vglm(y ~ 1, genbetaII, trace=TRUE)
    fit = vglm(y ~ 1, genbetaII(init.p=1.0, init.a=4, init.sc=7, init.q=2.3),
               trace=TRUE, crit="c")
    coef(fit, mat=TRUE)
    Coef(fit)
    summary(fit)

[Package VGAM version 0.7-9 Index]