VGAM (version 0.9-4)

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

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.

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.

See Also

Lino, genbetaII.

Examples

ldata <- data.frame(y1 = rbeta(n = 1000, exp(0.5), exp(1)))  # ~ standard beta
fit <- vglm(y1 ~ 1, lino, ldata, trace = TRUE)
coef(fit, matrix = TRUE)
Coef(fit)
head(fitted(fit))
summary(fit)

# Nonstandard beta distribution
ldata <-
  transform(ldata, y2 = rlino(n = 1000, shape1 = 2, shape2 = 3, lambda = exp(1)))
fit2 <- vglm(y2 ~ 1, lino(lshape1 = identitylink,
                          lshape2 = identitylink, ilamb = 10),
             data = ldata)
coef(fit2, matrix = TRUE)