If the observations are drawn from a continuous distribution (no ties in the sample),
the function ECBC() returns the commonly used empirical checkerboard copula.
If there are ties in the sample, the empirical copula is adjusted and calculated in the following way:
Let (u_i,v_i) := (F_n(x_i),G_n(y_i)) be the pseudo-observations for i in {1,...,n} and (u_1',v_1'),..., (u_m',v_m') the distinct pairs of pseudo-observations with m leq n. Moreover set S_1:={0, u_1, ..., u_m_1} and S_2:={0, v_1,..., v_m_2} and define the quantities t_i,r_i and s_i for i=1,..., m by
$$t_i := sum_{j=1}^n 1_{(u_i',v_i')}(u_j,v_j)$$
$$r_i := sum_{j=1}^n 1_{u_i}(u_j)$$
$$s_i := sum_{j=1}^n 1_{v_i}(v_j)$$
where 1 defines the indicator function.
Define the empirical subcopula A'_n: S_1 x S_2 to {0,1/n, ..., (n-1)/n,1} by
$$A'_n(s_1,s_2)= 1/n sum_{i=1}^m t_i * 1_{[0,s_1] x [0,s_2]} (u_i', v_i')=1/n sum_{i=1}^n 1_{[0,s_1] x [0,s_2]} (u_i, v_i)$$
for all s_1 in S_1 and s_2 in S_2.
We extend the subcopula A'_n to a copula by defining the transformations w_i:[0,1]^2 to [u_i'-r_i/n,u_i'] x [v_i'-s_i/n,v_i'] by
$$w_i(x,y)=(u_i'-r_i/n+r_i*x/n, v_i'-s_i/n + s_iy/n)$$
and set the measure of the empirical copula mu_A_n^B := 1/n sum_i=1^m t_i mu_B^w_i, where B denotes the product copula.