Simultaneous normal-equation post-selection confidence regions in model-free linear regression

⚠️ Curated Limitations & Negative Results

• Coordinatewise Projections Versus Joint Parallelepiped Shape: Remark 4.6 (p. 20) and Section 4.5 (pp. 22-23, Equation 29) show that while R_hat_{n,M}^dagger is a 2|M|-constraint parallelepiped whose joint Lebesgue volume scales as O_p(sqrt(|M| log(p)/n))^{|M|}, projecting R_hat_{n,M}^dagger onto individual coordinates yields coordinatewise confidence intervals of width 2(C_n^Gamma(alpha) + C_n^Sigma(alpha) ||beta_hat_{n,M}||_1) ||Sigma_hat_n^{-1}(M) e_j||_1 = O_p(sqrt(|M| log(p)/n) ||beta_hat_{n,M}||_1), which can be wider than Berk et al. (2013) coordinatewise intervals when |M| is close to k and ||beta_hat_{n,M}||_1 grows with sqrt(|M|).
• Sparse-Eigenvalue Condition (A1)(k) for R_hat_{n,M}^dagger and Null-Testing Power Loss: While R_hat_{n,M} (Equation 13) is finite-sample valid for any n, p >= 1 without eigenvalue conditions, the plug-in parallelepiped R_hat_{n,M}^dagger (Equation 14) requires Condition (A1)(k): k D_n^Sigma = o_p(Lambda_n(k)) with Lambda_n(k) = min_{M in M_p(k)} lambda_min(Sigma_n(M)) > 0 (Theorem 4.2), and Remark 4.7 (p. 20) proves that 0 in R_hat_{n,M} implies 0 in R_hat_{n,M}^dagger, so R_hat_{n,M}^dagger has strictly lower power than R_hat_{n,M} for testing H_{0,M}: beta_{n,M} = 0.
• Primary-Source Typographical Notes in arXiv:1806.04119v1: Verified five typographical slips in the primary manuscript: (1) Section 2.2 (p. 7, last line) prints 'beta_{n,M'}(1) != beta_{n,M'}(1)' where the second term should be 'beta_{n,M}(1)'; (2) the proofs of Theorem 4.1 (p. 17, Equation 18), Theorem 4.2 (p. 19, penultimate display), and Proposition 4.1 (Appendix B, p. 39, Equation 40) print population 'Sigma_n(M)' in place of empirical 'Sigma_hat_n(M)'; (3) Section 4.5 (p. 22, Equation 22) prints '(log p)^{2/alpha} (log n)^{2/alpha - 1/2} = o(n^{1/2})' with 'alpha' in the exponent instead of the sub-Weibull tail parameter 'gamma'; (4) Section 5 (p. 29, Step 2 of the multiplier bootstrap algorithm) prints '1/n sum_{i=1}^n e_{i,j}(W_i - W_bar_n)' instead of '1/sqrt(n) sum_{i=1}^n e_{i,j}(W_i - W_bar_n)' (matching S_n^{eW} on p. 28) and Step 3 sums 'i=1' to B_n while indexing 'j'; and (5) Section 6 (p. 30, line 13) prints 'Gamma_hat_n^star, Gamma_hat^star' for 'Gamma_hat_n^star, Gamma_n^star' and Theorem 6.1 (p. 31) prints 'k D_{2n}^star' for 'k D_n^{Sigma star}'.
• All 49 pages of the full-text PDF of Kuchibhotla, Brown, Buja, Cai, George, & Zhao (2020, arXiv:1806.04119v1, DOI:10.1214/19-AOS1917) were read. Constructed numerical checks in research/discovery/simultaneous-normal-equation-posi-confidence-regions-boundaries.py verify (1) Theorem 3.1 exact set-theoretic equivalence of universal post-selection coverage and simultaneous family-wise coverage across all 2^p - 1 = 63 submodels using the adversarial selector, plus the Equations (17)-(18) deterministic normal-equation inequality across all 18,900 (trial, submodel) pairs, (2) Theorems 4.1 & 4.2 finite-sample and asymptotic simultaneous coverage of R_hat_{n,M} and R_hat_{n,M}^dagger and Lemma 4.1 uniform L1 bound, (3) Remark 4.7 superior null-hypothesis testing power of R_hat_{n,M} over R_hat_{n,M}^dagger and Proposition 4.1 model-cardinality-adaptive volume vs. Berk et al. (2013) max-|t| volume, and (4) Section 4.6 exact tightness under fixed covariates for the greedy marginal selector and Section 5 polynomial-time O(n p^2) Gaussian multiplier bootstrap simultaneous coverage (96.0%).