The Two-parameter Beta Distribution Family Function

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

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.

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.

Author

Thomas W. Yee

Note

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

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")
#> Iteration 1: coefficients = 0.94886618, 2.81704941
#> Iteration 2: coefficients = 0.95450916, 2.83087867
#> Iteration 3: coefficients = 0.95454583, 2.83096779
#> Iteration 4: coefficients = 0.95454583, 2.83096779
fit <- vglm(y ~ 1, betaR, data = bdata, trace = TRUE, crit = "coef")
#> Iteration 1: coefficients = -0.050861872,  1.035706702
#> Iteration 2: coefficients = -0.046531622,  1.040605914
#> Iteration 3: coefficients = -0.046519624,  1.040618628
#> Iteration 4: coefficients = -0.046519624,  1.040618628
coef(fit, matrix = TRUE)
#>             loglink(shape1) loglink(shape2)
#> (Intercept)     -0.04651962        1.040619
Coef(fit)  # Useful for intercept-only models
#>    shape1    shape2 
#> 0.9545458 2.8309678 

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)
#> Iteration 1: loglikelihood = -1659.1546
#> Iteration 2: loglikelihood = -1659.1478
#> Iteration 3: loglikelihood = -1659.1478
Coef(fit)
#>    shape1    shape2 
#> 0.9545458 2.8309678 
c(meanY = with(bdata, mean(Y)), head(fitted(fit),2))
#>    meanY                   
#> 7.025732 7.017260 7.017260