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 \(i \in \{1,\ldots,n\}\) and \((u_1',v_1'),\ldots, (u_m',v_m')\) the distinct pairs of pseudo-observations with m leq n. Moreover set \(S_1:=\{0, u_1, \ldots, u_{m_1}\}\) and \(S_2:=\{0, v_1,\ldots, v_{m_2}\}\) and define the quantities \(t_i,r_i\) and
\(s_i\) for \(i=1,\ldots, m\) 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 \(A'_n: S_1 x S_2 \to \{0,1/n, \ldots, (n-1)/n,1\}\) 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.