# betaR: The Two-parameter Beta Distribution Family Function In VGAM: Vector Generalized Linear and Additive Models

## Description

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

## Usage

 ```1 2 3``` ```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) * (B-y)^(shape2-1) / [Beta(shape1,shape2) * (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) * shape1 / (shape1 + shape2), and these are the fitted values of the object.

For the standard beta distribution the variance of Y is shape1 * shape2 / ((1+shape1+shape2) * (shape1+shape2)^2). If σ^2= 1 / (1+shape1+shape2) then the variance of Y can be written mu*(1-mu)*sigma^2 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`.

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.

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

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11``` ```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)) ```

### Example output

```Loading required package: stats4
VGLM    linear loop  1 :  coefficients = 0.99453982, 2.66042022
VGLM    linear loop  2 :  coefficients = 1.0100043, 2.6876497
VGLM    linear loop  3 :  coefficients = 1.0102765, 2.6881580
VGLM    linear loop  4 :  coefficients = 1.0102766, 2.6881582
VGLM    linear loop  1 :  coefficients = 0.00023757603, 0.97934360189
VGLM    linear loop  2 :  coefficients = 0.010165579, 0.988799243
VGLM    linear loop  3 :  coefficients = 0.010224105, 0.988856275
VGLM    linear loop  4 :  coefficients = 0.010224107, 0.988856277
VGLM    linear loop  5 :  coefficients = 0.010224107, 0.988856277
loge(shape1) loge(shape2)
(Intercept)   0.01022411    0.9888563
shape1   shape2
1.010277 2.688158
VGLM    linear loop  1 :  loglikelihood = -1723.9357
VGLM    linear loop  2 :  loglikelihood = -1723.9035
VGLM    linear loop  3 :  loglikelihood = -1723.9035
shape1   shape2
1.010277 2.688158
meanY
7.181428 7.185306 7.185306
```

VGAM documentation built on Jan. 16, 2021, 5:21 p.m.