VGAM (version 1.0-4)

# bifrankcop: Frank's Bivariate Distribution Family Function

## Description

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

## Usage

bifrankcop(lapar = "loge", 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.

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

rbifrankcop, bifgmcop, simulate.vlm.

## Examples

Run this code
# NOT RUN {
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)