# bigumbelIexp: Gumbel's Type I Bivariate Distribution Family Function In VGAM: Vector Generalized Linear and Additive Models

 bigumbelIexp 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

```bigumbelIexp(lapar = "identitylink", iapar = NULL, imethod = 1)
```

### Arguments

 `lapar` Link function applied to the association parameter alpha. See `Links` for more choices. `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 `imethod`. `imethod` 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.

T. W. Yee

### References

Gumbel, E. J. (1960). Bivariate Exponential Distributions. Journal of the American Statistical Association, 55, 698–707.

`bifgmexp`.

### Examples

```nn <- 1000
gdata <- data.frame(y1 = rexp(nn), y2 = rexp(nn))
## Not run:  with(gdata, plot(cbind(y1, y2)))
fit <- vglm(cbind(y1, y2) ~ 1, bigumbelIexp, gdata, trace = TRUE)
coef(fit, matrix = TRUE)
Coef(fit)