frank {VGAM}R Documentation

Frank's Bivariate Distribution Family Function

Description

Estimate the correlation parameter of Frank's bivariate distribution using maximum likelihood estimation.

Usage

frank(lcorp="loge", icorp=2)

Arguments

lcorp Link function applied to the (positive) correlation parameter alpha. See Links for more choices.
icorp Numeric. Initial value for alpha. If a convergence failure occurs try assigning a different value.

Details

The cumulative distribution function is

P(Y1 <= y1, Y2 <= y2) = H_{alpha}(y1,y2) = log_{alpha} [1 + (alpha^(y1)-1)*(alpha^(y2)-1)/ (alpha-1)]

for alpha != 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 y2=y1 and y2=1-y1. When alpha>1 then h_{1/alpha}(1-y_1,y_2).

If alpha=1 then H(y1,y2)=y1*y2, i.e., uniform on the unit square. As alpha approaches 0 then H(y1,y2)=min(y1,y2). As alpha approaches infinity then H(y1,y2)=max(0,y1+y2-1).

A variant of Newton-Raphson is used, which only seems to work for an intercept model.

Value

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

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.

Author(s)

T. W. Yee

References

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

See Also

rfrank.

Examples

ymat = rfrank(n=2000, alpha=exp(4))
## Not run: plot(ymat)
fit = vglm(ymat ~ 1, fam=frank, trace=TRUE)
coef(fit, matrix=TRUE)
Coef(fit)
vcov(fit)
fitted(fit)[1:5,]
summary(fit)

[Package VGAM version 0.7-1 Index]