VGAM (version 0.7-1)

betaff: The Two-parameter Beta Distribution Family Function

Description

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

Usage

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

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.

earg

List. Extra argument associated with link containing any extra information. See Links for general information about VGAM link functions.

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=1, shape2=3)
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,]