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

 paralogistic R Documentation

## Paralogistic Distribution Family Function

### Description

Maximum likelihood estimation of the 2-parameter paralogistic distribution.

### Usage

``````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 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 \geq 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`.

T. W. Yee

### References

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

`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)
``````

VGAM documentation built on Sept. 19, 2023, 9:06 a.m.