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.

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.

See Also

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)
head(fitted(fit))
summary(fit)
# }

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