Fig. 3

Bias, MSE and size of the three p-value alternatives for \( {\widehat{er}}_{up\;}^{*} \). The number of simulation repetitions was a hundred thousand with \( \pi \) varying from 0 to 0.7 and the group size and weigh choices as indicated (left panel: \( {N}_0={N}_1=20 \)and\( {w}_0={w}_1=0.5 \) and right panel: \( {N}_0=31;{N}_1=17 \)and\( {w}_0=0.35;{w}_1=0.65 \)). The lines represent the results for the three p-value (i) \( {p}_0 \) in dark blue; (ii) \( {p}_{max} \) in purple; and (iii) \( {p}_{obs} \) in light blue. The top row depicts the bias, while the middle row depicts the MSE. The red line in the size graphs (bottom row) corresponds to \( {p}_{\pi} \). The size was calculated for a significance level of 10%