Fig. 10.

The features are multinomial as in Fig. 9 and the setting is otherwise the same as in Fig. 8. (A) Empirical cdf of $\mathrm{\Phi }\left({\widehat{\beta }}_j/{\sigma }_{\star }\right)$ for a null variable (${\beta }_j=0$). (B) P values given by the LLR approximation Eq. 11 for this same null variable. (C) Empirical distribution of the P values from B. (D) Same as C but displaying accuracy in the extreme. These results are based on $\mathrm{500,\; 000}$ replicates.