Frank's Bivariate Distribution Family Function
Estimate the association parameter of Frank's bivariate distribution by maximum likelihood estimation.
bifrankcop(lapar = "loglink", iapar = 2, nsimEIM = 250)
lapar |
Link function applied to the (positive) association parameter
alpha.
See |
iapar |
Numeric. Initial value for alpha. If a convergence failure occurs try assigning a different value. |
nsimEIM |
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
rbifrankcop.
For intercept-only models an alternative is to set nsimEIM=NULL
so that a variant of Newton-Raphson is used.
An object of class "vglmff" (see vglmff-class).
The object is used by modelling functions such as vglm
and vgam.
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.
T. W. Yee
Genest, C. (1987). Frank's family of bivariate distributions. Biometrika, 74, 549–555.
## 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) ## End(Not run)
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