biplackettcop {VGAM} | R Documentation |
Plackett's Bivariate Copula Family Function
Description
Estimate the association parameter of Plackett's bivariate distribution (copula) by maximum likelihood estimation.
Usage
biplackettcop(link = "loglink", ioratio = NULL, imethod = 1,
nsimEIM = 200)
Arguments
link |
Link function applied to the (positive) odds ratio |
ioratio |
Numeric. Optional initial value for |
imethod , nsimEIM |
Details
The defining equation is
\psi = H \times (1-y_1-y_2+H) / ((y_1-H) \times (y_2-H))
where
P(Y_1 \leq y_1, Y_2 \leq y_2) = H_{\psi}(y_1,y_2)
is the cumulative distribution function.
The density function is h_{\psi}(y_1,y_2) =
\psi [1 + (\psi-1)(y_1 + y_2 - 2 y_1 y_2) ] / \left(
[1 + (\psi-1)(y_1 + y_2) ]^2 - 4 \psi
(\psi-1) y_1 y_2 \right)^{3/2}
for \psi > 0
.
Some writers call \psi
the cross product ratio
but it is called the odds ratio here.
The support of the function is the unit square.
The marginal distributions here are the standard uniform although
it is commonly generalized to other distributions.
If \psi = 1
then
h_{\psi}(y_1,y_2) = y_1 y_2
,
i.e., independence.
As the odds ratio tends to infinity one has y_1=y_2
.
As the odds ratio tends to 0 one has y_2=1-y_1
.
Fisher scoring is implemented using rbiplackcop
.
Convergence is often quite slow.
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 2-column matrix with 0.5 values because the marginal distributions correspond to a standard uniform distribution.
Author(s)
T. W. Yee
References
Plackett, R. L. (1965). A class of bivariate distributions. Journal of the American Statistical Association, 60, 516–522.
See Also
Examples
## Not run:
ymat <- rbiplackcop(n = 2000, oratio = exp(2))
plot(ymat, col = "blue")
fit <- vglm(ymat ~ 1, fam = biplackettcop, trace = TRUE)
coef(fit, matrix = TRUE)
Coef(fit)
vcov(fit)
head(fitted(fit))
summary(fit)
## End(Not run)