sinmad {VGAM} | R Documentation

## Singh-Maddala Distribution Family Function

### Description

Maximum likelihood estimation of the 3-parameter Singh-Maddala distribution.

### Usage

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

### Arguments

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

### Details

The 3-parameter Singh-Maddala distribution is the 4-parameter generalized beta II distribution with shape parameter p=1. It is known under various other names, such as the Burr XII (or just the Burr distribution), Pareto IV, beta-P, and generalized log-logistic distribution. More details can be found in Kleiber and Kotz (2003).

Some distributions which are special cases of the 3-parameter Singh-Maddala are the Lomax (a=1), Fisk (q=1), and paralogistic (a=q).

The Singh-Maddala distribution has density

f(y) = aq y^(a-1) / [b^a (1 + (y/b)^a)^(1+q)]

for a > 0, b > 0, q > 0, y > 0. Here, b is the scale parameter `scale`, and the others are shape parameters. The cumulative distribution function is

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

The mean is

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

provided -a < 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`.

### 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

`Sinmad`, `genbetaII`, `betaII`, `dagum`, `fisk`, `invlomax`, `lomax`, `paralogistic`, `invparalogistic`.

### Examples

    y = rsinmad(n=3000, 3, 5, 2)
    fit = vglm(y ~ 1, sinmad, trace=TRUE)
    fit = vglm(y ~ 1, sinmad, trace=TRUE, crit="c")
    coef(fit, mat=TRUE)
    Coef(fit)
    summary(fit)

[Package VGAM version 0.7-9 Index]