gumbelIbiv {VGAM} | R Documentation

## Gumbel's Type I Bivariate Distribution Family Function

### Description

Estimate the association parameter of Gumbel's Type I bivariate distribution by maximum likelihood estimation.

### Usage

    gumbelIbiv(lapar="identity", earg=list(), iapar=NULL, method.init=1)

### Arguments

`lapar` | Link function applied to the association parameter alpha. See `Links` for more choices.
`earg` | List. Extra argument for the link. See `earg` in `Links` for general information.
`iapar` | Numeric. Optional initial value for alpha. By default, an initial value is chosen internally. If a convergence failure occurs try assigning a different value. Assigning a value will override the argument `method.init`.
`method.init` | An integer with value `1` or `2` which specifies the initialization method. If failure to converge occurs try the other value, or else specify a value for `ia`.

### Details

The cumulative distribution function is

P(Y1 <= y1, Y2 <= y2) = exp(-y1-y2+alpha*y1*y2) + 1 - exp(-y1) - exp(-y2)

for real alpha. The support of the function is for y1>0 and y2>0. The marginal distributions are an exponential distribution with unit mean.

A variant of Newton-Raphson is used, which only seems to work for an intercept model. It is a very good idea to set `trace=TRUE`.

### 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. Currently, the fitted value is a matrix with two columns and values equal to 1. This is because each marginal distribution corresponds to a exponential distribution with unit mean.

This VGAM family function should be used with caution.

### Author(s)

T. W. Yee

### References

Castillo, E., Hadi, A. S., Balakrishnan, N. Sarabia, J. S. (2005) Extreme Value and Related Models with Applications in Engineering and Science, Hoboken, N.J.: Wiley-Interscience.

### See Also

`morgenstern`.

### Examples

    n = 1000
    ymat = cbind(rexp(n), rexp(n))
    ## Not run: plot(ymat)
    fit = vglm(ymat ~ 1, fam=gumbelIbiv, trace=TRUE)
    fit = vglm(ymat ~ 1, fam=gumbelIbiv, trace=TRUE, crit="coef")
    coef(fit, matrix=TRUE)
    Coef(fit)
    head(fitted(fit))

[Package VGAM version 0.7-9 Index]