Error in TLL CDF integrate

[biennie2002]

When calling the TllBicop in bicop, there seems to be an error in pbicop().

true_copula <- bicop_dist(family = "gaussian", parameters = 0.5)
x <- rbicop(1000, true_copula)
fit_x <- bicop(x, family_set = "tll")

xy <- expand.grid(u1 = seq(0, 1, length.out = 5),
                  u2 = seq(0, 1, length.out = 5))
zz <- pbicop(as.matrix(xy), fit_x)

contour(x = seq(0, 1, length.out = 5),
        y = seq(0, 1, length.out = 5),
        z = matrix(zz, nrow = 5, byrow = FALSE))

The CDF surface seems to have some issue.

I found the possible issue in tools_interpolation.ipp: InterpolationGrid::integrate_2d()

tmpint  = int_on_grid(upr, tmpvals2, grid_points_);
tmpint1 = int_on_grid(1.0, tmpvals2, grid_points_);
return std::min(std::max(tmpint / tmpint1, 1e-10), 1 - 1e-10);

It seems that dividing by tmpint1 (which is approximately u2) scales the CDF incorrectly. Removing the division fixes the issue:

tmpint = int_on_grid(upr, tmpvals2, grid_points_);
return std::min(std::max(tmpint, 1e-10), 1.0 - 1e-10);