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

 betaII R Documentation

## Beta Distribution of the Second Kind

### Description

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

### Usage

``````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 \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.

### 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`.

### Examples

``````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)
``````

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