# betaII: Beta Distribution of the Second Kind In VGAM: Vector Generalized Linear and Additive Models

## Description

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

## Usage

 ```1 2 3 4 5``` ```betaII(lscale = "loglink", lshape2.p = "loglink", lshape3.q = "loglink", 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.

## 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 >= 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.

## 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.

`betaff`, `genbetaII`, `dagum`, `sinmad`, `fisk`, `inv.lomax`, `lomax`, `paralogistic`, `inv.paralogistic`.
 ```1 2 3 4 5 6 7 8``` ```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) ```