paralogistic {VGAM} | R Documentation

## Paralogistic Distribution Family Function

### Description

Maximum likelihood estimation of the 2-parameter paralogistic distribution.

### Usage

    paralogistic(lscale = "loge", lshape1.a = "loge", iscale = NULL,
        ishape1.a = NULL, imethod = 1, lss = TRUE, gscale = exp(-5:5),
        gshape1.a = seq(0.75, 4, by = 0.25), probs.y = c(0.25, 0.5, 0.75),
        zero = "shape")

### Arguments

`lss` | See `CommonVGAMffArguments` for important information.
`lshape1.a, lscale` | Parameter link functions applied to the (positive) parameters a and `scale`. See `Links` for more choices.
`iscale, ishape1.a, imethod, zero` | See `CommonVGAMffArguments` for information. For `imethod = 2` a good initial value for `ishape1.a` is needed to obtain good estimates for the other parameter.
`gscale, gshape1.a` | See `CommonVGAMffArguments` for information.
`probs.y` | See `CommonVGAMffArguments` for information.

### 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; these are returned as the fitted values. This family function handles multiple responses.

### Value

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

### Note

See the notes in `genbetaII`.

### Author(s)

T. W. Yee

### References

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

### See Also

`Paralogistic`, `sinmad`, `genbetaII`, `betaII`, `dagum`, `fisk`, `inv.lomax`, `lomax`, `inv.paralogistic`.

### Examples

    pdata <- data.frame(y = rparalogistic(n = 3000, exp(1), scale = exp(1)))
    fit <- vglm(y ~ 1, paralogistic(lss = FALSE), data = pdata, trace = TRUE)
    fit <- vglm(y ~ 1, paralogistic(ishape1.a = 2.3, iscale = 5),
                data = pdata, trace = TRUE)
    coef(fit, matrix = TRUE)
    Coef(fit)
    summary(fit)

[Package VGAM version 1.0-3 Index]