gammahyp {VGAM} | R Documentation

## Gamma Hyperbola Bivariate Distribution

### Description

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

### Usage

    gammahyp(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).

### 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.

### Author(s)

T. W. Yee

### References

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

### See Also

`exponential`.

### Examples

    x  = runif(n <- 1000)
    theta = exp(-2+x)
    y1 = rexp(n, rate=exp(-theta)/theta)
    y2 = 1 + rexp(n, rate=theta)
    fit = vglm(cbind(y1,y2) ~ x, fam=gammahyp(expected=TRUE), trace=TRUE)
    fit = vglm(cbind(y1,y2) ~ x, fam=gammahyp, trace=TRUE, crit="coef")
    coef(fit, matrix=TRUE)
    Coef(fit)
    head(fitted(fit))
    summary(fit)

[Package VGAM version 0.7-9 Index]