Simes closed-testing concentration-set shortcut and double-asymptotic false discovery proportion consistency

Simes closed-testing concentration-set shortcut and double-asymptotic false discovery proportion (FDP) consistency is the multiple-testing methodology introduced by Goeman, Meijer, Krebs, and Solari (2019) for closed testing (Marcus, Peritz, & Gabriel 1976; Goeman & Solari 2011) with Simes local tests I in U_alpha iff min_{1 <= i <= |I|} |I| * p_{(i:I)} / i <= alpha across m hypotheses H_1, ..., H_m with true null index set T subseteq {1, ..., m}. Letting K_i = {r_{m-i+1}, ..., r_m} denote the index set of the i largest p-values and defining Hommel's worst-case non-rejected intersection size h_alpha = max{0 <= i <= m : K_i not in U_alpha} (which yields the simultaneous (1 - alpha)-upper confidence bound pi_hat_alpha = h_alpha / m on |T| / m), the method characterizes closed-testing rejections X_alpha = {I : forall J supseteq I, J in U_alpha} via I in X_alpha iff exists 1 <= i <= |I| such that h_alpha * p_{(i:I)} <= i * alpha (Lemma 2), and establishes the exact all-subset shortcut (Theorem 1) for the simultaneous (1 - alpha)-lower bound d_alpha(S) = |S| - t_alpha(S) on true discoveries |S \ T| and upper bound q_alpha(S) = t_alpha(S) / |S| on the FDP pi(S) = |T cap S| / |S|: d_alpha(S) = max_{1 <= u <= |S|} (1 - u + |{i in S : h_alpha * p_i <= u * alpha}|), where the maximization over u restricts to 1 <= u <= z_alpha - m + h_alpha + 1 with z_alpha = min{m - h_alpha <= i <= m : h_alpha * p_{(i)} <= (i - m + h_alpha + 1) * alpha} for h_alpha < m and 0 for h_alpha = m (Lemma 3), enabling O(|S|) per-subset evaluation after a one-time O(m log m) computation of h_alpha. The concentration set Z_alpha = L_{z_alpha} of the z_alpha smallest p-values is the unique minimal set satisfying d_alpha(S) = d_alpha(S cap Z_alpha) for all S subseteq {1, ..., m}, with d_alpha(Z_alpha) = m - h_alpha and |S cap R_alpha| <= d_alpha(S) <= |S cap Z_alpha| <= |S cap B_alpha| between Hommel's elementary FWER set R_alpha and the Benjamini-Hochberg (BH) set B_alpha = L_{b_alpha} (Lemma 4), while alpha * q_alpha(B_q) <= pi_hat_alpha * q holds simultaneously for all q in [0, 1] (Lemma 5). Under Efron's mixture model P_n(x) = gamma * x + (1 - gamma) * P_{1,n}(x), Non-consonant Simes rejections yield non-vanishing average discovery power whenever P_n is Simes-detectable (bar_pi_alpha = inf_{0 <= x < 1} (1 - P_n(x * alpha)) / (1 - x) < 1, Lemma 6 and Theorem 2) even though |R_alpha| / m ->^P 0, and for any subset sequence S_m with plim_{m -> infty} |S_m| / m = c > 0 and consistent alternative tests lim_{n -> infty} P^S_{1,n}(x) = 1, q_{alpha,m,n}(S_m) is a doubly asymptotically consistent estimator of the true subset FDP gamma_S = plim_{m -> infty} pi(S_m) as (m, n) -> infty at arbitrary relative rates for any fixed alpha in (0, 1) (Theorem 3).