VGAM (version 1.0-4)

betaR: The Two-parameter Beta Distribution Family Function

Description

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

Usage

betaR(lshape1 = "loge", lshape2 = "loge",
      i1 = NULL, i2 = NULL, trim = 0.05,
      A = 0, B = 1, parallel = FALSE, zero = NULL)

Arguments

lshape1, lshape2, i1, i2

Details at CommonVGAMffArguments. See Links for more choices.

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 outliers.

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.

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.

See Also

betaff, Beta, genbetaII, betaII, betabinomialff, betageometric, betaprime, rbetageom, rbetanorm, kumar, simulate.vlm.

Examples

# NOT RUN {
bdata <- data.frame(y = rbeta(n = 1000, shape1 = exp(0), shape2 = exp(1)))
fit <- vglm(y ~ 1, betaR(lshape1 = "identitylink", lshape2 = "identitylink"),
            data = bdata, trace = TRUE, crit = "coef")
fit <- vglm(y ~ 1, betaR, data = bdata, trace = TRUE, crit = "coef")
coef(fit, matrix = TRUE)
Coef(fit)  # Useful for intercept-only models

bdata <- transform(bdata, Y = 5 + 8 * y)  # From 5 to 13, not 0 to 1
fit <- vglm(Y ~ 1, betaR(A = 5, B = 13), data = bdata, trace = TRUE)
Coef(fit)
c(meanY = with(bdata, mean(Y)), head(fitted(fit),2))
# }