VGAM (version 1.0-2)

gammahyperbola: Gamma Hyperbola Bivariate Distribution

Description

Estimate the parameter of a gamma hyperbola bivariate distribution by maximum likelihood estimation.

Usage

gammahyperbola(ltheta = "loge", itheta = NULL, expected = FALSE)

Arguments

ltheta

Link function applied to the (positive) parameter $theta$. See Links for more choices.

itheta

Initial value for the parameter. The default is to estimate it internally.

expected

Logical. FALSE means the Newton-Raphson (using the observed information matrix) algorithm, otherwise the expected information matrix is used (Fisher scoring algorithm).

Value

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

Details

The joint probability density function is given by $$f(y_1,y_2) = \exp( -e^{-\theta} y_1 / \theta - \theta y_2 )$$ for $theta > 0$, $y1 > 0$, $y2 > 1$. The random variables $Y1$ and $Y2$ are independent. The marginal distribution of $Y1$ is an exponential distribution with rate parameter $exp(-theta)/theta$. The marginal distribution of $Y2$ is an exponential distribution that has been shifted to the right by 1 and with rate parameter $theta$. The fitted values are stored in a two-column matrix with the marginal means, which are $theta * exp(theta)$ and $1 + 1/theta$.

The default algorithm is Newton-Raphson because Fisher scoring tends to be much slower for this distribution.

References

Reid, N. (2003) Asymptotics and the theory of inference. Annals of Statistics, 31, 1695--1731.

See Also

exponential.

Examples

gdata <- data.frame(x2 = runif(nn <- 1000))
gdata <- transform(gdata, theta = exp(-2 + x2))
gdata <- transform(gdata, y1 = rexp(nn, rate = exp(-theta)/theta),
                          y2 = rexp(nn, rate = theta) + 1)
fit <- vglm(cbind(y1, y2) ~ x2, gammahyperbola(expected = TRUE), data = gdata)
coef(fit, matrix = TRUE)
Coef(fit)
head(fitted(fit))
summary(fit)