VGAM (version 0.9-4)

bigumbelI: 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

bigumbelI(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

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

See Also

morgenstern.

Examples

nn <- 1000
gdata <- data.frame(y1 = rexp(nn), y2 = rexp(nn))
with(gdata, plot(cbind(y1, y2)))
fit <- vglm(cbind(y1, y2) ~ 1, fam = bigumbelI, data = gdata, trace = TRUE)
coef(fit, matrix = TRUE)
Coef(fit)
head(fitted(fit))