VGAM (version 1.0-4)

# betaII: Beta Distribution of the Second Kind

## Description

Maximum likelihood estimation of the 3-parameter beta II distribution.

## Usage

```betaII(lscale = "loge", lshape2.p = "loge", lshape3.q = "loge",
iscale = NULL, ishape2.p = NULL, ishape3.q = NULL, imethod = 1,
gscale = exp(-5:5), gshape2.p = exp(-5:5),
gshape3.q = seq(0.75, 4, by = 0.25),
probs.y = c(0.25, 0.5, 0.75), zero = "shape")```

## Arguments

lscale, lshape2.p, lshape3.q

Parameter link functions applied to the (positive) parameters `scale`, `p` and `q`. See `Links` for more choices.

iscale, ishape2.p, ishape3.q, imethod, zero

See `CommonVGAMffArguments` for information.

gscale, gshape2.p, gshape3.q

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 3-parameter beta II is the 4-parameter generalized beta II distribution with shape parameter \(a=1\). It is also known as the Pearson VI distribution. Other distributions which are special cases of the 3-parameter beta II include the Lomax (\(p=1\)) and inverse Lomax (\(q=1\)). More details can be found in Kleiber and Kotz (2003).

The beta II distribution has density \$\$f(y) = y^{p-1} / [b^p B(p,q) \{1 + y/b\}^{p+q}]\$\$ for \(b > 0\), \(p > 0\), \(q > 0\), \(y \geq 0\). Here, \(b\) is the scale parameter `scale`, and the others are shape parameters. The mean is \$\$E(Y) = b \, \Gamma(p + 1) \, \Gamma(q - 1) / (\Gamma(p) \, \Gamma(q))\$\$ provided \(q > 1\); 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.

`betaff`, `genbetaII`, `dagum`, `sinmad`, `fisk`, `inv.lomax`, `lomax`, `paralogistic`, `inv.paralogistic`.

## Examples

Run this code
``````# NOT RUN {
bdata <- data.frame(y = rsinmad(2000, shape1.a = 1, shape3.q = exp(2),
scale = exp(1)))  # Not genuine data!
fit <- vglm(y ~ 1, betaII, data = bdata, trace = TRUE)
fit <- vglm(y ~ 1, betaII(ishape2.p = 0.7, ishape3.q = 0.7),
data = bdata, trace = TRUE)
coef(fit, matrix = TRUE)
Coef(fit)
summary(fit)
# }
``````

Run the code above in your browser using DataCamp Workspace