paralogistic {VGAM} | R Documentation

## Paralogistic Distribution Family Function

### Description

Maximum likelihood estimation of the 2-parameter paralogistic distribution.

### Usage

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

### Arguments

`link.a, link.scale` | Parameter link functions applied to the (positive) shape parameter `a` and (positive) scale parameter `scale`. See `Links` for more choices.
`earg.a, earg.scale` | 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`.
`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} which correspond to `a`, `scale`, respectively.

### Details

The 2-parameter paralogistic distribution is the 4-parameter generalized beta II distribution with shape parameter p=1 and a=q. It is the 3-parameter Singh-Maddala distribution with a=q. More details can be found in Kleiber and Kotz (2003).

The 2-parameter paralogistic has density

f(y) = a^2 y^(a-1) / [b^a (1 + (y/b)^a)^(1+a)]

for a > 0, b > 0, y > 0. Here, b is the scale parameter `scale`, and a is the shape parameter. The mean is

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

provided a > 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

`Paralogistic`, `genbetaII`, `betaII`, `dagum`, `fisk`, `invlomax`, `lomax`, `invparalogistic`.

### Examples

    y = rparalogistic(n=3000, 4, 6)
    fit = vglm(y ~ 1, paralogistic, trace=TRUE)
    fit = vglm(y ~ 1, paralogistic(init.a=2.3, init.sc=5),
               trace=TRUE, crit="c")
    coef(fit, mat=TRUE)
    Coef(fit)
    summary(fit)

[Package VGAM version 0.7-9 Index]