VGAM (version 1.0-4)

# paralogistic: 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.

## Value

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

## 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.

## 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

Run this code
```# NOT RUN {
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)
# }
```

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