VGAM (version 1.1-8)

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 = "loglink", lshape2 = "loglink", llambda = "loglink",
     ishape1 = NULL,   ishape2 = NULL,   ilambda = 1, zero = NULL)

Value

An object of class "vglmff"

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

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.

Author

T. W. Yee

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 loglink 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)))  # Std beta
fit <- vglm(y1 ~ 1, lino, data = ldata, trace = TRUE)
coef(fit, matrix = TRUE)
Coef(fit)
head(fitted(fit))
summary(fit)

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