Compute the expected value of \(U\) given a \(V\) (the \(Y\) direction) through the conditional distribution function \(G(Y)\) using the appropriate partial derivative of a copula (\(\mathbf{C}(u,v)\)) with respect to \(V\). The inversion of the partial derivative is the conditional quantile function. Basic principles provide the expectation for a \(y \ge 0\) is
$$E[Y] = \int_0^\infty yf(y)\mathrm{d}y = \int_0^\infty \bigl(1-G_y(Y)\bigr)\mathrm{d}y\mbox{,}$$
which for the setting here becomes
$$E[U \mid V = v] = \int_0^1 \bigl(1 - \frac{\delta}{\delta v} \mathbf{C}(u,v)\bigr)\mathrm{d}u\mbox{.}$$
This function solves the integral using the derCOP2 function. This avoids a call of the derCOPinv2 through its uniroot() inversion of the derivative. The example shown for EuvCOP() below does a validation check using conditional simulation, which is dependence (of course) of the design of the copBasic package, as part of simple isolation of a horizontal slice of the simulation and computing the mean of the \(V\) within the slice.
EuvCOP(v=seq(0.01, 0.99, by=0.01), cop=NULL, para=NULL, asuv=FALSE, nsim=1E5,
subdivisions=100L, rel.tol=.Machine$double.eps^0.25, abs.tol=rel.tol, ...)Value(s) for the expectation are returned.
Nonexceedance probability \(v\) in the \(Y\) direction;
A copula function with vectorization as in asCOP;
Vector of parameters or other data structures, if needed, to pass to the copula;
Return a data frame of the \(U\) and \(V\);
Number of simulations for Monte Carlo integration when the numerical integration fails (see Note);
Argument of same name passed to integrate();
Argument of same name passed to integrate();
Argument of same name passed to integrate(); and
Additional arguments to pass to derCOP2.
W.H. Asquith
EvuCOP, derCOP2