The Beta-Prime Distribution
Estimation of the two shape parameters of the beta-prime distribution by maximum likelihood estimation.
betaprime(lshape = "loglink", ishape1 = 2, ishape2 = NULL, zero = NULL)
lshape |
Parameter link function applied to the two (positive) shape parameters.
See |
ishape1, ishape2, zero |
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 Y1 has a gamma(shape1) distribution and Y2 has a gamma(shape2) distribution then Y1/Y2 has a Betaprime(shape1,shape2) distribution.
An object of class "vglmff" (see vglmff-class).
The object is used by modelling functions such as vglm,
rrvglm
and vgam.
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.
Thomas W. Yee
Johnson, N. L. and Kotz, S. and Balakrishnan, N. (1995). Chapter 25 of: Continuous Univariate Distributions, 2nd edition, Volume 2, New York: Wiley.
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 modelsPlease choose more modern alternatives, such as Google Chrome or Mozilla Firefox.