frank {VGAM}R Documentation

Frank's Bivariate Distribution Family Function

Description

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

Usage

frank(lapar="loge", eapar=list(), iapar=2, nsimEIM=250)

Arguments

lapar Link function applied to the (positive) association parameter alpha. See Links for more choices.
eapar List. Extra argument for the link. See earg in Links for general information.
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(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).

The default is to use Fisher scoring implemented using rfrank. 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.

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, fgm.

Examples

ymat = rfrank(n=2000, alpha=exp(4))
## Not run: plot(ymat, col="blue")
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 0.7-7 Index]