betaR: The Two-parameter Beta Distribution Family Function

In VGAM: Vector Generalized Linear and Additive Models

The Two-parameter Beta Distribution Family Function

Description

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

Usage

betaR(lshape1 = "loglink", lshape2 = "loglink",
      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.

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.

Value

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

Note

The response must have values in the interval (A, B). VGAM 0.7-4 and prior called this function betaff.

Author(s)

Thomas W. Yee

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

bdata <- data.frame(y = rbeta(1000, shape1 = exp(0), shape2 = exp(1)))
fit <- vglm(y ~ 1, betaR(lshape1 = "identitylink",
            lshape2 = "identitylink"), 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))

VGAM documentation built on Sept. 18, 2024, 9:09 a.m.