VGAM (version 1.1-13)

bifrankcop: Frank's Bivariate Distribution Family Function

Description

Estimate the association parameter of Frank's bivariate distribution by maximum likelihood estimation.

Usage

bifrankcop(lapar = "loglink", iapar = 2, nsimEIM = 250)

Value

An object of class "vglmff"

(see vglmff-class). The object is used by modelling functions such as vglm

and vgam.

Arguments

lapar

Link function applied to the (positive) association parameter $\alpha$. See Links for more choices.

iapar

Numeric. Initial value for $\alpha$. If a convergence failure occurs try assigning a different value.

nsimEIM

See CommonVGAMffArguments.

Author

T. W. Yee

Details

The cumulative distribution function is $$P(Y_1\le y_1,Y_2\le y_2)=H_{\alpha }(y_1,y_2)={\log }_{\alpha }[1+({\alpha }^{y_1}-1)({\alpha }^{y_2}-1)/(\alpha -1)]$$ for $\alpha \ne 1$. Note the logarithm here is to base $\alpha$. The support of the function is the unit square.

When $0<\alpha <1$ the probability density function $h_{\alpha }(y_1,y_2)$ is symmetric with respect to the lines $y_2=y_1$ and $y_2=1-y_1$. When $\alpha >1$ then $h_{\alpha }(y_1,y_2)=h_{1/\alpha }(1-y_1,y_2)$.

$\alpha =1$ then $H(y_1,y_2)=y_1{y}_2$, i.e., uniform on the unit square. As $\alpha$ approaches 0 then $H(y_1,y_2)=min(y_1,y_2)$. As $\alpha$ approaches infinity then $H(y_1,y_2)=max(0,y_1+y_2-1)$.

The default is to use Fisher scoring implemented using rbifrankcop. For intercept-only models an alternative is to set nsimEIM=NULL so that a variant of Newton-Raphson is used.

References

Genest, C. (1987). Frank's family of bivariate distributions. Biometrika, 74, 549--555.

See Also

rbifrankcop, bifgmcop, simulate.vlm.

Examples

if (FALSE) {
ymat <- rbifrankcop(n = 2000, apar = exp(4))
plot(ymat, col = "blue")
fit <- vglm(ymat ~ 1, fam = bifrankcop, trace = TRUE)
coef(fit, matrix = TRUE)
Coef(fit)
vcov(fit)
head(fitted(fit))
summary(fit)
}