get small, grows with $d$. Consequently, we expect that many of the $\bar{p}_r$ should be close to $\alpha=0.05$. On the other hand, if $d$ is not large then many matrices $R$ will lead to a biased rejection probability due to misspecification of the working model. This is exactly what we observe in Figures \[fig:Boxplots1\] and \[fig:Boxplots2\]. For small values of $d$, the rejection probabilities $\bar{p}_r$ are systematically biased and we see some variability of their values due to the variation in the choice of $R_r$ (compare benchmark panel in Figure \[fig:Boxplots2\]). Both the bias and the variability in $\bar{p}_r$ reduce when $d$ increases, which is what we expected, as for large $d$, most $R_r$ will be favorable and we obtain small misspecification errors uniformly over these favorable $R_r$. What is remarkable is the systematic over-rejection in case of the $t$- and exponential distribution and the under-rejection for Bernoulli and uniformly distributed designs. We currently can not explain the mechanism that is responsible for this pattern. Finally, the benchmark panel shows i.i.d. samples $(\tilde{p}_r)_{r=1}^{100}$ with $\tilde{p}_r \thicksim \text{Binomial}(1000,\alpha)/1000$ and success probabilities $\alpha = 0.05, 0.1, 0.15,0.2$. This provides some idea what portion of the variability observed in the other panels is 