VGAM (version 1.0-4)

# bigumbelIexp: 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.

## Value

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

## Details

The cumulative distribution function is $$P(Y_1 \leq y_1, Y_2 \leq y_2) = e^{-y_1-y_2+\alpha y_1 y_2} + 1 - e^{-y_1} - e^{-y_2}$$ for real $$\alpha$$. The support of the function is for $$y_1>0$$ and $$y_2>0$$. 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.

## References

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

bifgmexp.

## Examples

Run this code
# NOT RUN {
nn <- 1000
gdata <- data.frame(y1 = rexp(nn), y2 = rexp(nn))
# }
# NOT RUN {
with(gdata, plot(cbind(y1, y2)))
# }
# NOT RUN {
fit <- vglm(cbind(y1, y2) ~ 1, bigumbelIexp, data = gdata, trace = TRUE)
coef(fit, matrix = TRUE)
Coef(fit)