betaprime: 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

Value

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

Details

The beta-prime distribution is given by

f (y) = y^{s h a p e 1 - 1} (1 + y)^{- s h a p e 1 - s h a p e 2} / B (s h a p e 1, s h a p e 2)

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 $Y_1$ has a $gamma(shape1)$ distribution and $Y_2$ has a $gamma(shape2)$ distribution then $Y_1/Y_2$ has a $Betaprime(shape1,shape2)$ distribution.

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.

Examples

Run this code

nn = 1000
betadat = data.frame(shape1 = exp(1), shape2 = exp(3))
betadat = transform(betadat, yb = rbeta(nn, shape1, shape2))
betadat = transform(betadat, y1 = (1-yb)/yb, y2 = yb/(1-yb),
                             y3 = rgamma(nn, exp(3)) / rgamma(nn, exp(2)))

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

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

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

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

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