VGAM (version 0.8-7)

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, imethod=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 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.

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, NJ, USA: 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)
head(fitted(fit))