VGAM (version 0.7-3)

betaff: The Two-parameter Beta Distribution Family Function

Description

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

Usage

betaff(link = "loge", earg=list(),
       i1 = NULL, i2 = NULL, trim = 0.05,
       A = 0, B = 1, zero = NULL)

Arguments

link

Parameter link function applied to the two shape parameters. See Links for more choices. A log link (default) ensures that the parameters are positive.

earg

List. Extra argument for the link. 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.

zero

An integer-valued vector specifying which linear/additive predictors are modelled as intercepts only. If used, the value must be from the set {1,2} which correspond to the first and second shape parameters respectively.

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 $Y$.

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

See Also

Beta, genbetaII, betaII, betabin.ab, betageometric, betaprime, rbetageom, rbetanorm, beta4.

Examples

y = rbeta(n=1000, shape1=exp(0), shape2=exp(1))
fit = vglm(y ~ 1, betaff(link="identity"), trace = TRUE, crit="c")
fit = vglm(y ~ 1, betaff, trace = TRUE, crit="c")
coef(fit, matrix=TRUE)
Coef(fit)  # Useful for intercept-only models

Y = 5 + 8 * y    # From 5 to 13, not 0 to 1
fit = vglm(Y ~ 1, betaff(A=5, B=13), trace = TRUE)
Coef(fit)  
fitted(fit)[1:4,]