Finite- and Large-Sample Inference for Ranks using Multinomial Data with an Application to Ranking P

An inspection of the proof of Theorem 2.1 reveals that the confidence set Rn not only covers our definition of the rank, rj0 , in the sense of (1), but also any other “reasonable” definition of the rank in the sense that { } P R contj0 ⊆ Rn ≥ 1− α, where Rcontn ≡ [min(Rn),max(Rn)] is the interval from the smallest to the largest value in the confidence set Rn. [...] Therefore, the validity of the bootstrap in the sense of (11) implies that the bootstrap confidence set for the ranks also covers the true ranks with probability at least 1− α in the limit as n→∞. [...] The first column shows that Clopper-Pearson confidence sets are strictly wider than the exact Holm confidence sets in 3.1% of category×territory cases, the non-studentized bootstrap confidence sets are strictly wider than the exact Holm in 29.2% of cases, and the studentized bootstrap confidence sets are strictly wider than the exact Holm confidence sets in 47.7% of cases. [...] bootStud confidence sets are very wide at the bottom of the ranking Below we explore each of these findings in depth by focusing on lengths and empirical coverage 23 frequencies of confidence sets for the rank of the 1st (top of the ranking), 4th (middle of the ranking) and 7th (bottom of the ranking) categories in Table 4. [...] While not necessarily covering the true values of the differences, the bootstrap confidence sets for the differences lead to the correct determination of their signs and thus to coverage of the rank.