Frank's Bivariate Distribution Family Function

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

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

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.

Details

The cumulative distribution function is $$P(Y_1 \leq y_1, Y_2 \leq y_2) = H_{\alpha}(y_1,y_2) = \log_{\alpha} [1 + (\alpha^{y_1}-1)(\alpha^{y_2}-1)/ (\alpha-1)] $$ 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.

Value

An object of class "vglmff" (see vglmff-class). The object is used by modelling functions such as vglm and vgam.

References

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

Author

T. W. Yee

Note

The response must be a two-column matrix. Currently, the fitted value is a matrix with two columns and values equal to a half. This is because the marginal distributions correspond to a standard uniform distribution.

Examples

if (FALSE) { # \dontrun{
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)
} # }