VGAM (version 0.8-4.1)

beta.ab: The Two-parameter Beta Distribution Family Function

Description

Estimation of the shape parameters of the two-parameter beta distribution.

Usage

beta.ab(lshape1 = "loge", lshape2 = "loge",
        eshape1 = list(), eshape2 = list(),
        i1 = NULL, i2 = NULL, trim = 0.05,
        A = 0, B = 1, parallel = FALSE, zero = NULL)

Arguments

lshape1, lshape2

Parameter link functions applied to the two shape parameters. See Links for more choices. The log link (defaults) ensures that the parameters are positive.

eshape1, eshape2

List. Extra argument for the links. See earg in Links for general information.

i1, i2

Initial value for the first and second shape parameters respectively. A NULL value means it is obtained in the initialize slot.

trim

An argument which is fed into mean(); it is the fraction (0 to 0.5) of observations to be trimmed from each end of the response y before the mean is computed. This is used when computing initial values, and guards against o

A, B

Lower and upper limits of the distribution. The defaults correspond to the standard beta distribution where the response lies between 0 and 1.

parallel, zero

See CommonVGAMffArguments for more information.

Value

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

Details

The two-parameter beta distribution is given by $f(y) =$ $$(y-A)^{shape1-1} \times (B-y)^{shape2-1} / [Beta(shape1,shape2) \times (B-A)^{shape1+shape2-1}]$$ for $A < y < B$, and $Beta(.,.)$ is the beta function (see beta). The shape parameters are positive, and here, the limits $A$ and $B$ are known. The mean of $Y$ is $E(Y) = A + (B-A) \times shape1 / (shape1 + shape2)$, and these are the fitted values of the object.

For the standard beta distribution the variance of $Y$ is $shape1 \times shape2 / [(1+shape1+shape2) \times (shape1+shape2)^2]$. If $\sigma^2= 1 / (1+shape1+shape2)$ then the variance of $Y$ can be written $\sigma^2 \mu (1-\mu)$ where $\mu=shape1 / (shape1 + shape2)$ is the mean of $Y$.

Another parameterization of the beta distribution involving the mean and a precision parameter is implemented in betaff.

If $A$ and $B$ are unknown, then the VGAM family function beta4() can be used to estimate these too.

References

Johnson, N. L. and Kotz, S. and Balakrishnan, N. (1995) Chapter 25 of: Continuous Univariate Distributions, 2nd edition, Volume 2, New York: Wiley.

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

Documentation accompanying the VGAM package at http://www.stat.auckland.ac.nz/~yee contains further information and examples.

See Also

betaff, Beta, genbetaII, betaII, betabinomial.ab, betageometric, betaprime, rbetageom, rbetanorm, kumar, beta4.

Examples

betadat = data.frame(y = rbeta(n=1000, shape1=exp(0), shape2=exp(1)))
fit = vglm(y ~ 1, beta.ab(lshape1="identity", lshape2="identity"),
           data=betadat, trace=TRUE, crit="c")
fit = vglm(y ~ 1, beta.ab, data=betadat, trace=TRUE, crit="c")
coef(fit, matrix=TRUE)
Coef(fit)  # Useful for intercept-only models

betadat = transform(betadat, Y = 5 + 8 * y)   # From 5 to 13, not 0 to 1
fit = vglm(Y ~ 1, beta.ab(A=5, B=13), data=betadat, trace=TRUE)
Coef(fit)
c(meanY=with(betadat, mean(Y)), head(fitted(fit),2))