VGAM (version 0.7-8)

fgm: Farlie-Gumbel-Morgenstern's Bivariate Distribution Family Function

Description

Estimate the association parameter of Farlie-Gumbel-Morgenstern's bivariate distribution by maximum likelihood estimation.

Usage

fgm(lapar="rhobit", earg=list(), iapar=NULL,
    method.init=1, nsimEIM=200)

Arguments

lapar

Link function applied to the association parameter $\alpha$, which is real. 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.

nsimEIM

See CommonVGAMffArguments for more information.

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 $$P(Y_1 \leq y_1, Y_2 \leq y_2) = y_1 y_2 ( 1 + \alpha (1 - y_1) (1 - y_2) )$$ for $-1 < \alpha < 1$. The support of the function is the unit square. The marginal distributions are the standard uniform distributions. When $\alpha = 0$ the random variables are independent.

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

rfgm, frank, morgenstern.

Examples

ymat = rfgm(n = 1000, alpha=rhobit(3, inverse=TRUE))
plot(ymat, col="blue")
fit = vglm(ymat ~ 1, fam=fgm, trace=TRUE)
coef(fit, matrix=TRUE)
Coef(fit)
fitted(fit)[1:5,]