Estimate the association parameter of Plackett's bivariate distribution (copula) by maximum likelihood estimation.
Link function applied to the (positive) odds ratio psi.
Numeric. Optional initial value for psi. If a convergence failure occurs try assigning a value or a different value.
The defining equation is
psi = H*(1-y1-y2+H) / ((y1-H)*(y2-H))
where P(Y1 <= y1, Y2 <= y2)= H(y1,y2) is the cumulative distribution function. The density function is h(y1,y2) =
psi*[1 + (psi-1)*(y1 + y2 - 2*y1*y2) ] / ( [1 + (psi-1)*(y1 + y2)]^2 - 4*psi*(psi-1)*y1*y2)^(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(y1,y2) = y1*y2, i.e., independence. As the odds ratio tends to infinity one has y1=y2. As the odds ratio tends to 0 one has y2=1-y1.
Fisher scoring is implemented using
Convergence is often quite slow.
An object of class
The object is used by modelling functions such as
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.
T. W. Yee
Plackett, R. L. (1965). A class of bivariate distributions. Journal of the American Statistical Association, 60, 516–522.
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