VGAM (version 1.0-4)

# lino: Generalized Beta Distribution Family Function

## Description

Maximum likelihood estimation of the 3-parameter generalized beta distribution as proposed by Libby and Novick (1982).

## Usage

lino(lshape1 = "loge", lshape2 = "loge", llambda = "loge",
ishape1 = NULL,   ishape2 = NULL,   ilambda = 1, zero = NULL)

## Arguments

lshape1, lshape2

Parameter link functions applied to the two (positive) shape parameters $$a$$ and $$b$$. See Links for more choices.

llambda

Parameter link function applied to the parameter $$\lambda$$. See Links for more choices.

ishape1, ishape2, ilambda

Initial values for the parameters. A NULL value means one is computed internally. The argument ilambda must be numeric, and the default corresponds to a standard beta distribution.

zero

Can be an integer-valued vector specifying which linear/additive predictors are modelled as intercepts only. Here, the values must be from the set {1,2,3} which correspond to $$a$$, $$b$$, $$\lambda$$, respectively. See CommonVGAMffArguments for more information.

## Value

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

## Details

Proposed by Libby and Novick (1982), this distribution has density $$f(y;a,b,\lambda) = \frac{\lambda^{a} y^{a-1} (1-y)^{b-1}}{ B(a,b) \{1 - (1-\lambda) y\}^{a+b}}$$ for $$a > 0$$, $$b > 0$$, $$\lambda > 0$$, $$0 < y < 1$$. Here $$B$$ is the beta function (see beta). The mean is a complicated function involving the Gauss hypergeometric function. If $$X$$ has a lino distribution with parameters shape1, shape2, lambda, then $$Y=\lambda X/(1-(1-\lambda)X)$$ has a standard beta distribution with parameters shape1, shape2.

Since $$\log(\lambda)=0$$ corresponds to the standard beta distribution, a summary of the fitted model performs a t-test for whether the data belongs to a standard beta distribution (provided the loge link for $$\lambda$$ is used; this is the default).

## References

Libby, D. L. and Novick, M. R. (1982) Multivariate generalized beta distributions with applications to utility assessment. Journal of Educational Statistics, 7, 271--294.

Gupta, A. K. and Nadarajah, S. (2004) Handbook of Beta Distribution and Its Applications, NY: Marcel Dekker, Inc.

Lino, genbetaII.

## Examples

Run this code
# NOT RUN {
ldata <- data.frame(y1 = rbeta(n = 1000, exp(0.5), exp(1)))  # ~ standard beta
fit <- vglm(y1 ~ 1, lino, data = ldata, trace = TRUE)
coef(fit, matrix = TRUE)
Coef(fit)