gumbelIbiv: 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, imethod=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 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} + α 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, N.J.: Wiley-Interscience.

Examples

Run this code

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

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