betaprime: The Beta-Prime Distribution

In VGAM: Vector Generalized Linear and Additive Models

The Beta-Prime Distribution

Description

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

Usage

betaprime(lshape = "loglink", ishape1 = 2, ishape2 = NULL,
          zero = NULL)

Arguments

lshape	Parameter link function applied to the two (positive) shape parameters. See Links for more choices.
ishape1, ishape2, zero	See CommonVGAMffArguments for more information.

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; these are returned as the fitted values.

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.

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.

See Also

betaff, Beta.

Examples

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

fit1 <- vglm(y1 ~ 1, betaprime, data = bdata, trace = TRUE)
coef(fit1, matrix = TRUE)

fit2 <- vglm(y2 ~ 1, betaprime, data = bdata, trace = TRUE)
coef(fit2, matrix = TRUE)

fit3 <- vglm(y3 ~ 1, betaprime, data = bdata, trace = TRUE)
coef(fit3, matrix = TRUE)

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

VGAM documentation built on Sept. 18, 2024, 9:09 a.m.