VGAM (version 0.7-10)

frank: Frank's Bivariate Distribution Family Function

Description

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

Usage

frank(lapar="loge", eapar=list(), iapar=2, nsimEIM=250)

Arguments

lapar

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

eapar

List. Extra argument for the link. See earg in Links for general information.

iapar

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

nsimEIM

See CommonVGAMffArguments.

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 $$P(Y_1\le y_1,Y_2\le y_2)=H_{\alpha }(y_1,y_2)={\log }_{\alpha }[1+({\alpha }^{y_1}-1)({\alpha }^{y_2}-1)/(\alpha -1)]$$ 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 $h_{\alpha}(y_1,y_2) = h_{1/\alpha}(1-y_1,y_2)$.

If $\alpha=1$ 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 rfrank. 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

rfrank, fgm.

Examples

ymat = rfrank(n=2000, alpha=exp(4))
plot(ymat, col="blue")
fit = vglm(ymat ~ 1, fam=frank, trace=TRUE)
coef(fit, matrix=TRUE)
Coef(fit)
vcov(fit)
head(fitted(fit))
summary(fit)