# gammahyperbola: Gamma Hyperbola Bivariate Distribution In VGAM: Vector Generalized Linear and Additive Models

## Description

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

## Usage

 `1` ```gammahyperbola(ltheta = "loglink", 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).

## Details

The joint probability density function is given by

f(y1,y2) = exp( -exp(-theta) * y1 / theta - theta * y2)

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.

## Value

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

## Note

The response must be a two-column matrix.

T. W. Yee

## References

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

`exponential`.
 ```1 2 3 4 5 6 7 8 9``` ```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) ```