# inv.paralogistic: Inverse Paralogistic Distribution Family Function In VGAM: Vector Generalized Linear and Additive Models

## Description

Maximum likelihood estimation of the 2-parameter inverse paralogistic distribution.

## Usage

 ```1 2 3 4``` ```inv.paralogistic(lscale = "loglink", lshape1.a = "loglink", 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 a good estimate for the other parameter. `gscale, gshape1.a` See `CommonVGAMffArguments` for information. `probs.y` See `CommonVGAMffArguments` for information.

## Details

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

The inverse paralogistic distribution has density

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

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(a + 1/a) gamma(1 - 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`.

T. W. Yee

## References

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

`Inv.paralogistic`, `genbetaII`, `betaII`, `dagum`, `sinmad`, `fisk`, `inv.lomax`, `lomax`, `paralogistic`, `simulate.vlm`.
 ```1 2 3 4 5 6 7``` ```idata <- data.frame(y = rinv.paralogistic(n = 3000, exp(1), scale = exp(2))) fit <- vglm(y ~ 1, inv.paralogistic(lss = FALSE), data = idata, trace = TRUE) fit <- vglm(y ~ 1, inv.paralogistic(imethod = 2, ishape1.a = 4), data = idata, trace = TRUE, crit = "coef") coef(fit, matrix = TRUE) Coef(fit) summary(fit) ```