betaprime {VGAM} | R Documentation

## The Beta-Prime Distribution

### Description

Estimation of the two shape parameters of the beta-prime distribution by maximum likelihood estimation.

### Usage

    betaprime(link = "loge", earg=list(), i1 = 2, i2 = NULL, zero = NULL)

### Arguments

`link` | Parameter link function applied to the two (positive) shape parameters. See `Links` for more choices.
`earg` | List. Extra argument for each of the links. See `earg` in `Links` for general information.
`i1, i2` | Initial values for the first and second shape parameters. A `NULL` value means it is obtained in the `initialize` slot. Note that `i2` is obtained using `i1`.
`zero` | An integer-valued vector specifying which linear/additive predictors are modelled as intercepts only. The value must be from the set {1,2} corresponding respectively to `shape1` and `shape2` respectively. If `zero=NULL` then both parameters are modelled with the explanatory variables.

### Details

The beta-prime distribution is given by

f(y) = y^(shape1-1) * (1+y)^(-shape1-shape2) / B(shape1,shape2)

for y > 0. The shape parameters are positive, and here, B is the beta function. The mean of Y is shape1 / (shape2-1) provided shape2>1.

If Y has a Beta(shape1,shape2) distribution then Y/(1-Y) and (1-Y)/Y have a Betaprime(shape1,shape2) and Betaprime(shape2,shape1) distribution respectively. Also, if Y1 has a gamma(shape1) distribution and Y2 has a gamma(shape2) distribution then Y1/Y2 has a Betaprime(shape1,shape2) distribution.

### 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 positive values only.

The beta-prime distribution is also known as the beta distribution of the second kind or the inverted beta distribution.

### 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.

Documentation accompanying the VGAM package at http://www.stat.auckland.ac.nz/~yee contains further information and examples.

### See Also

`betaff`.

### Examples

    yb = rbeta(n <- 1000, shape1=exp(1), shape2=exp(3))
    y1 = (1-yb)/yb
    y2 = yb/(1-yb)
    y3 = rgamma(n, exp(3)) / rgamma(n, exp(2))

    fit1 = vglm(y1 ~ 1, betaprime, trace=TRUE)
    coef(fit1, matrix=TRUE)

    fit2 = vglm(y2 ~ 1, betaprime, trace=TRUE)
    coef(fit2, matrix=TRUE)

    fit3 = vglm(y3 ~ 1, betaprime, trace=TRUE)
    coef(fit3, matrix=TRUE)

    # Compare the fitted values
    mean(y3)
    head(fitted(fit3))
    Coef(fit3)  # Useful for intercept-only models

[Package VGAM version 0.7-9 Index]