VGAM (version 1.1-9)

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

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)
}