betaII {VGAM} | R Documentation

## Beta Distribution of the Second Kind

### Description

Maximum likelihood estimation of the 3-parameter beta II distribution.

### Usage

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

### Arguments

`link.scale, link.p, link.q` | Parameter link functions applied to the (positive) parameters `scale`, `p` and `q`. See `Links` for more choices.
`earg.scale, earg.p, earg.q` | List. Extra argument for each of the links. See `earg` in `Links` for general information.
`init.scale, init.p, init.q` | Optional initial values for `scale`, `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} which correspond to `scale`, `p`, `q`, respectively.

### Details

The 3-parameter beta II is the 4-parameter generalized beta II distribution with shape parameter a=1. It is also known as the Pearson VI distribution. Other distributions which are special cases of the 3-parameter beta II include the Lomax (p=1) and inverse Lomax (q=1). More details can be found in Kleiber and Kotz (2003).

The beta II distribution has density

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

for b > 0, p > 0, q > 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) gamma(q - 1) / ( gamma(p) gamma(q))

provided q > 1.

### 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`.

### Author(s)

T. W. Yee

### References

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

### See Also

`betaff`, `genbetaII`, `dagum`, `sinmad`, `fisk`, `invlomax`, `lomax`, `paralogistic`, `invparalogistic`.

### Examples

    y = rsinmad(n=2000, a=1, 6, 2)  # Not genuine data!
    fit = vglm(y ~ 1, betaII, trace=TRUE)
    fit = vglm(y ~ 1, betaII(init.p=0.7, init.q=0.7), trace=TRUE, crit="c")
    coef(fit, mat=TRUE)
    Coef(fit)
    summary(fit)

[Package VGAM version 0.7-9 Index]