# Table 8 Performances of the second approximation $$\left( {\tilde{r}_{{q_{\text{c}} ,\hat{q}_{\text{c}} }}^{2} } \right)$$ when the parents of the candidates belong to the reference population as a function of the number of markers (n M ) and reference population size (n R ), assuming ν2 = 0.4

n M n R True value $$\left( {{r}_{{q_{\text{c}} ,\hat{q}_{\text{c}} }}^{2} } \right)$$ Approximation $$\left( {\tilde{r}_{{q_{\text{c}} ,\hat{q}_{\text{c}} }}^{2} } \right)$$ 10th order approximation $${\text{E}}\left[ {\tilde{r}_{{q_{\text{c}} ,\hat{q}_{\text{c}} }}^{2} } \right]$$
1000 500 0.37 0.46 0.35 0.46
1000 1000 0.53 0.60 0.54 0.61
1000 1500 0.64 0.70 0.65 0.69
1000 2000 0.71 0.75 0.72 0.75
1500 500 0.30 0.39 0.26 0.40
1500 1000 0.47 0.55 0.46 0.51
1500 1500 0.56 0.63 0.56 0.61
1500 2000 0.63 0.69 0.64 0.68
2000 500 0.27 0.36 0.22 0.35
2000 1000 0.40 0.49 0.38 0.48
2000 1500 0.50 0.58 0.50 0.56
2000 2000 0.57 0.64 0.57 0.62
2500 500 0.24 0.33 0.20 0.32
2500 1000 0.34 0.44 0.31 0.45
2500 1500 0.44 0.53 0.43 0.53
2500 2000 0.37 0.46 0.35 0.46
1. The convergence criterion is the value of the Taylor series at order 10
2. $${\text{E}}\left[ {\tilde{r}_{{q_{\text{c}} ,\hat{q}_{\text{c}} }}^{2} } \right]$$ is the expectation of the second approximation across the distribution of allele frequencies as given in Goddard 