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.

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

## Examples

Run this code
# 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))
# }


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