{"id":"5b0bfd85-c376-4327-a2b9-a3e49566d63a","arxiv_id":"2501.07645","paper_version":2,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":4.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The leading coverage error of Student's t intervals is (z/n)[(z^2-3)/6 κ - (z^4+2z^2-3)/9 γ^2] φ(z), the normal-theory expression with no constant.","lead":"This note derives the missing asymptotic coverage error formula for the ordinary Student's t confidence interval, filling a gap in a well-known table from Hall (1988). The resulting formula is the same as the normal-theory interval's formula, but with the constant term removed.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; the t-interval coverage error formula follows cleanly from Hall's expansion, with only minor presentational issues.","rationale":"The reader's verdict is ACCEPT with high confidence and low correctness risk. My stress-test pass confirms the algebraic logic: the t interval is transformed into a normal-theory interval with a level α' that differs from α by O(1/n); substituting Hall's expansion (4) at α' and adding the nominal-level correction 2(α−α') cancels the intercept and leaves the κ and γ² coefficients unchanged. This matches Hall's own remark that the −3.35 term arises from the difference between the normal and Student t distribution functions. The paper's cited reliance on Hall's regularity conditions is the natural weakest point, exactly as the reader stated, but it is a scope limitation rather than an internal inconsistency. The O(n^{-3/2}) remainder is honestly stated. The only concrete issue I found is a minor numerical typo in the Gaussian-data illustrative check, which does not touch the main result. Therefore the verdict should remain unchanged, with no material corrections needed to the central claim.","tokens_in":4363,"tokens_out":24500,"duration_ms":225598,"concrete_test":"Independently re-derive the two-sided coverage error for the Student t interval directly from a standard Edgeworth expansion of the Studentized mean (without the α' substitution), using the same regularity conditions as Hall (1988), and verify that the O(1/n) coefficient is exactly z_{1−α}[(z_α^2−3)/6 κ − (z_α^4+2z_α^2−3)/9 γ²]φ(z_α). If the direct derivation reproduces this coefficient, the central claim is fully confirmed. As a secondary check, recompute the Gaussian-data coverage error from equation (5) to confirm the value is −0.39/n rather than −0.78/n, resolving the typo.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central derivation is sound. The paper maps the Student t interval to the normal-theory interval at a shifted level α', uses Hall's one-sided Edgeworth expansion (4), and adds the level correction. The algebra is internally consistent: the shifted quantile and s/σ̂ correction yield α' = α + (z_α^3+3z_α)/(4n)φ(z_α), and the added term 2(α−α') exactly cancels the intercept that distinguishes the normal-theory entry from the t-entry. The resulting coefficient z_{1−α}[(z_α^2−3)/6 κ − (z_α^4+2z_α^2−3)/9 γ²]φ(z_α) matches Hall's row with the −3.35 term removed. The acknowledged reliance on Hall's regularity conditions is handled by explicit reference to Hall (1988), and the O(n^{-3/2}) remainder is stated rather than overclaimed. No load-bearing flaw in the argument was found; the manuscript even transparently retracts its earlier factor-of-two misreading. A minor typographical inconsistency exists in the illustrative Gaussian-data check: equation (5) and Hall's table imply a coverage error of approximately −0.39/n for the normal-theory interval, while the text says −0.78/n. This is a presentational slip that does not affect the derived formula for the Student t interval.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The note derives an asymptotic two-sided coverage-error formula for the usual Student t confidence interval for an IID sample mean, thereby filling a missing row in Table 1 of Hall (1988). The argument maps the t interval to the normal-theory interval at an adjusted level, using the Abramowitz and Stegun quantile expansion and Hall's one-sided Edgeworth expansion. The resulting coefficient is (z_{1-alpha}/n)[(z_alpha^2-3)/6 kappa - (z_alpha^4+2z_alpha^2-3)/9 gamma^2] phi(z_alpha) + O(n^{-3/2}), which for 95% intervals is the normal-theory entry 0.14 kappa - 2.12 gamma^2 - 3.35 with the intercept removed. The paper also transparently retracts an earlier factor-of-two correction claim.","tokens_in":4619,"tokens_out":14423,"duration_ms":125657,"significance":"If correct, the result fills a real gap: Student t intervals are ubiquitous in practice, yet Hall's table lists an entry for the normal-theory interval but not for the t interval. The derivation is algebraically clean, uses only external published formulas, and involves no fitted parameters or circular reasoning. The explicit statement that the remainder is O(n^{-3/2}), rather than the stronger O(n^{-2}) sometimes available for bootstrap methods, is appropriately cautious. The transparent withdrawal of the earlier factor-of-two claim is a strength. The main value is completing a standard reference table and providing a simple theoretical explanation of the good coverage behavior observed in RQMC applications. No load-bearing flaw was found in the central derivation.","major_comments":[],"minor_comments":[{"comment":"Equation (6) displays the intercept term as -z_{1-alpha}(z_alpha^2+3)/2, but to match Equation (5) and the rounded value -3.35 after multiplication by z_{1-alpha}, the denominator should be 4, not 2. As printed, the intercept evaluates to about -6.70 instead of -3.35.","section":"Section 2, Eq. (6)"},{"comment":"The text after Equation (5) states that the normal-theory interval has Gaussian-data coverage error approximately -0.78/n, but Equations (3) and (5) give approximately -0.39/n. The displayed value should be corrected, although it does not affect the derived Student t formula.","section":"Section 2, Gaussian-data check"},{"comment":"The abstract's formula 2 Phi^{-1}(0.975)(A kappa + gamma^2 + C) phi(1.96)/n is missing the coefficient B on gamma^2; it should read A kappa + B gamma^2 + C.","section":"Abstract"},{"comment":"The reliance on Hall (1988) for regularity conditions is acceptable, but the note should explicitly state the inherited conditions, such as finite moments and the validity of the Edgeworth expansion, so that the formula's domain is clear without requiring the reader to consult Hall's paper.","section":"Section 1"},{"comment":"The display after \"Now\" is garbled: 2(alpha-alpha') should be -(z_alpha^3+3z_alpha)/(2n) phi(z_alpha) + O(n^{-2}), which equals z_{1-alpha}(z_alpha^2+3)/(2n) phi(z_alpha) + O(n^{-2}); the printed version omits phi(z_alpha) and places an extra n in the denominator.","section":"Section 3, adjusted interval"}],"recommendation":"minor_revision","confidential_remarks":"This is a short, useful note that fills a small gap in a well-known table. The transparent retraction of the earlier factor-of-two correction is a point in its favor. I recommend minor revision to fix the typographical inconsistencies in the displayed equations and the Gaussian-data check."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This is a short note that does exactly what it says: it supplies the asymptotic two-sided coverage error formula for Student's t confidence intervals, which Hall's Table 1 omits. The punchline is that the t interval's formula is Hall's normal-theory entry with the constant term removed: 0.14κ − 2.12γ² instead of 0.14κ − 2.12γ² − 3.35. The derivation is clean and transparent: write the t interval as a normal-theory interval at an adjusted level α′, use Hall's one-sided Edgeworth expansion, add the level shift, and watch the intercept cancel. The algebra checks out. The paper also honestly retracts its earlier misreading of Hall's table, which is a good sign.\n\nThe strengths are real. The result genuinely completes a classic reference table, and the method is reusable. The paper states its limits: it inherits Hall's regularity conditions without restating them, and it only tracks terms to O(n^{−3/2}), not the O(n^{−2}) that bootstrap corrections sometimes achieve. Those are appropriate caveats, not flaws.\n\nThe one soft spot I found is a minor presentational slip in the Gaussian-data illustration. The text says the normal-theory interval has coverage error ≈ −0.78/n, but equation (5) and Hall's table both imply ≈ −0.39/n. The displayed formula in that sentence has an extra factor of two. This is a typo in an aside; it does not affect the main derivation or the final Student-t formula.\n\nWho is this for? Anyone who uses t intervals in nonparametric or simulation settings, especially the RQMC audience that motivated the note, and anyone teaching from Hall's table. It is a small but solid contribution, not a breakthrough. I would cite it if I were writing about coverage error comparisons. It deserves a serious referee: the math is correct, the exposition is clear, and the missing row is a legitimate gap in the literature. I would accept after a minor text correction.","headline":"A clean, correct note that fills the missing Student-t row in Hall's 1988 table; the derivation is sound and the only real issue is a minor arithmetic slip in the Gaussian check.","tokens_in":5046,"tokens_out":2833,"would_cite":true,"duration_ms":25722,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62E20","62F25","62G15"],"pacs":[],"model":"deepseek-v4-flash","headline":"For IID samples, the two-sided coverage error of the Student's t confidence interval is $\\frac{z_{1-\\alpha}}{n}\\left(\\frac{z_\\alpha^2-3}{6}\\kappa - \\frac{z_\\alpha^4+2z_\\alpha^2-3}{9}\\gamma^2\\right)\\varphi(z_\\alpha)+O(n^{-3/2})$, filling…","keywords":["Student's t","coverage error","Edgeworth expansion","skewness","excess kurtosis","confidence interval","asymptotic expansion","randomized quasi-Monte Carlo"],"falsifier":"Simulate the two-sided t interval for a distribution with known $\\gamma$ and $\\kappa$ (for example a shifted lognormal, or a symmetric mixture with $\\gamma=0$ and chosen $\\kappa$) at several sample sizes, and compare the fitted coefficient of $1/n$ to $z_{0.975}\\varphi(1.96)(0.14\\kappa - 2.12\\gamma^2)$; a mismatch that persists as $n$ grows under Hall's regularity conditions would refute the formula.","tokens_in":4177,"feed_emoji":"📊","tokens_out":10871,"duration_ms":95784,"temperature":0.7,"pith_summary":"This paper fills a missing entry in Hall's 1988 table of asymptotic coverage errors for nonparametric confidence intervals for the mean. The usual Student's t interval, the workhorse method, was absent from that table, and the paper derives its two-sided coverage error from Hall's own one-sided Edgeworth expansion. For an IID sample with standardized third moment $\\gamma$ and excess kurtosis $\\kappa$, the leading error is $(z_{1-\\alpha}/n)[(z_\\alpha^2-3)\\kappa/6 - (z_\\alpha^4+2z_\\alpha^2-3)\\gamma^2/9]\\varphi(z_\\alpha) + O(n^{-3/2})$. At the 95% level this rounds to $1.96(0.14\\kappa - 2.12\\gamma^2)\\varphi(1.96)/n$, exactly Hall's normal-theory entry with the $-3.35$ intercept removed. The result pins down when the standard interval over- or undercovers and explains why it can be surprisingly robust.","feed_headline":"Student t intervals finally get their missing coverage formula","feed_subtitle":"Hall's normal-table entry with the −3.35 intercept dropped; for Gaussian data the leading error vanishes.","key_machinery":"The argument runs through Hall's one-sided Edgeworth expansion (equation 4, page 949) for a Gaussian-quantile interval with the MLE variance $\\hat\\sigma^2$. The t interval is reached by two perturbative substitutions. First, the Student t quantile $t^\\alpha_{(n-1)}$ is expanded in terms of $z_\\alpha$ using the Abramowitz-Stegun expansion $t^\\alpha_{(n-1)} = z_\\alpha + (z_\\alpha^3+z_\\alpha)/(4n)+O(n^{-2})$. Second, the sample standard deviation replaces the MLE via $s = \\hat\\sigma\\sqrt{n/(n-1)} = \\hat\\sigma(1+1/(2n)+O(n^{-2}))$. Together these shift the effective normal quantile level by an amount that cancels the intercept in Hall's normal-theory formula, leaving only the $\\kappa$ and $\\gamma^2$ terms at order $1/n$.","core_discovery":"On the paper's own terms, the central discovery is that the Student's t interval's coverage error inherits the skewness and kurtosis terms of the Gaussian-quantile interval in Hall's Table 1, while the intercept term $-3.35$ disappears. The derived formula is $$\\text{coverage error} = \\frac{z_{1-\\$\\alpha$}}{n}\\left(\\frac{z_\\$alpha^{2}$-3}{6}\\kappa - \\frac{z_\\$alpha^{4}$+2z_\\$alpha^{2}$-3}{9}\\$gamma^{2}$\\right)\\varphi(z_\\$\\alpha$)+O($n^{{-3/2}}$).$$ At $\\alpha=0.025$ the constants round to $0.14$ for $\\kappa$ and $-2.12$ for $\\gamma^2$, and the $-3.35$ Gaussian intercept is exactly cancelled by the combined effect of Student's heavier-tailed quantiles and the unbiased variance estimator $s^2$. For Gaussian data, setting $\\gamma=\\kappa=0$ makes the leading error vanish, consistent with the exact finite-sample property that the t interval has correct coverage for normals.","pith_inferences":["A step the paper leaves implicit: because the intercept vanishes for every distribution, the t interval has no built-in undercoverage from the normal approximation; users who want conservative intervals should check the sign of $0.14\\kappa - 2.12\\gamma^2$, which may be either positive or negative.","A testable extension: the same quantile-and-variance substitution could be applied to the other entries in Hall's table that use Gaussian quantiles, producing a column of t-based coverage formulas; the intercept removal should hold in each case.","In the RQMC setting that motivated the paper, the t interval's observed robustness is consistent with the absence of the intercept: the remaining $O(1/n)$ term is small whenever the replicate distribution has low skewness and modest kurtosis, and the paper's formula gives the exact condition."],"forward_implications":["The 95% Student t interval's leading coverage error is proportional to $(0.14\\kappa - 2.12\\gamma^2)/n$, so its sign is governed by the balance between excess kurtosis and squared skewness; distributions with $0.14\\kappa \\approx 2.12\\gamma^2$ get near-nominal coverage to order $1/n$.","For Gaussian data the t interval has no leading $O(1/n)$ coverage error, while the normal-theory interval undercovers by roughly $0.78/n$; the t interval is therefore strictly better in the Gaussian reference case.","Because the $-3.35$ intercept is absent from every distribution, not just Gaussian ones, the t interval avoids a systematic normal-distribution bias that is independent of shape.","Hall's bootstrap comparison table can now be compared directly with the t interval on the same $\\gamma,\\kappa$ scale, since the missing entry is filled."],"supporting_citations":[{"why":"Supplies the one-sided Edgeworth expansion (equation 4) and the Table 1 entry whose missing t analog is derived.","marker":"Hall (1988)"},{"why":"Supplies the asymptotic expansion (7) connecting Student t quantiles to Gaussian quantiles, used to adjust the critical value.","marker":"Abramowitz and Stegun (1972)"},{"why":"Motivating application: documents the RQMC t-interval robustness that the missing formula is meant to explain; not used in the proof.","marker":"L'Ecuyer et al. (2023)"}],"fun_headline_variants":["Student t coverage error formula derived, Hall's table complete","t intervals' coverage error: no intercept, zero for Gaussian data","Hall's table missing entry for Student t now filled","Student t coverage error: Hall's -3.35 term vanishes"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The formula inherits Hall's one-sided Edgeworth expansion wholesale, so it holds only when the underlying distribution has the finite moments and smoothness Hall's regularity conditions require, and it only tracks terms through $O(1/n)$.","fun_headline_variants_meta":{"raw":{"variants":["Student t coverage error formula derived, Hall's table complete","t intervals' coverage error: no intercept, zero for Gaussian data","Hall's table missing entry for Student t now filled","Student t coverage error: Hall's -3.35 term vanishes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00113,"raw_usage":{"total_tokens":4753,"prompt_tokens":1061,"completion_tokens":3692,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":677,"completion_tokens_details":{"reasoning_tokens":3622}},"tokens_in":677,"tokens_out":3692,"duration_ms":25202,"temperature":1.0,"reasoning_tokens":3622,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T20:37:37.548337+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate the two-sided t interval for a distribution with known $\\gamma$ and $\\kappa$ (for example a shifted lognormal, or a symmetric mixture with $\\gamma=0$ and chosen $\\kappa$) at several sample sizes, and compare the fitted coefficient of $1/n$ to $z_{0.975}\\varphi(1.96)(0.14\\kappa - 2.12\\gamma^2)$; a mismatch that persists as $n$ grows under Hall's regularity conditions would refute the formula.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the one-sided Edgeworth expansion (equation 4) and the Table 1 entry whose missing t analog is derived."},{"cited_title":"and Stegun, I","cited_arxiv_id":null,"evidence_quote":"Supplies the asymptotic expansion (7) connecting Student t quantiles to Gaussian quantiles, used to adjust the critical value."},{"cited_title":"K., Owen, A","cited_arxiv_id":null,"evidence_quote":"Motivating application: documents the RQMC t-interval robustness that the missing formula is meant to explain; not used in the proof."}],"review_version":1}