Coverage errors for Student's t confidence intervals comparable to those in Hall (1988)

Art B. Owen

Coverage errors for Student's t confidence intervals comparable to those in Hall (1988)

T0 review · 0 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2501.07645 v2 pith:3W4ITQGI submitted 2025-01-13 math.ST cs.NAmath.NAstat.TH

Art B. Owen This is my paper · ORCID

classification math.STcs.NAmath.NAstat.TH MSC 62E2062F2562G15

keywords Student'stcoverageerrorEdgeworthexpansionskewnessexcesskurtosisconfidenceintervalasymptoticrandomizedquasi-MonteCarlo

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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 $1/n$.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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 $O(1/n)$.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Formalized claims in Lean

Claim #1: 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}\$gamm

/-- @claim 1 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}\$gamm -/
def central_claim : Prop :=

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

The 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.

The 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.

Who 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.

Referee Report

0 major / 5 minor

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.

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.

minor comments (5)

[Section 2, Eq. (6)] 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 2, Gaussian-data check] 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.
[Abstract] 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 1] 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 3, adjusted interval] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the t-interval coverage error formula is derived from external published expansions, not from the claimed result itself.

full rationale

The paper's central derivation maps the Student-t interval to a normal-theory interval at an adjusted level alpha', then applies Hall's one-sided Edgeworth expansion (equation (4), from Hall 1988, page 949) and Abramowitz and Stegun's asymptotic t-quantile expansion (equation (7)). The final formula 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}) is obtained by explicit algebra: the shifted quantile and s/sigma-hat correction produce alpha' = alpha + (z_alpha^3+3z_alpha)/(4n)phi(z_alpha), and the added term 2(alpha-alpha') exactly cancels the intercept term from Hall's normal-theory entry. No parameter is fitted to the target formula, and the claimed result is not used as an input to its own derivation. The only self-citation (L'Ecuyer et al. 2023, on which Owen is a coauthor) appears as motivation for revisiting the problem, not as a load-bearing premise. The paper also transparently corrects its own earlier factor-of-two misreading. The derivation is therefore self-contained with respect to the cited external expansions, and no circular step is present.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

No free parameters are introduced. The coefficients 0.14 and 2.12 are derived from the normal quantile z=1.96, not fitted. The two imported building blocks are Hall's one-sided coverage expansion and Abramowitz and Stegun's t quantile expansion; both are external and published.

assumptions (2)

domain assumption Hall's one-sided Edgeworth expansion for the normal-theory interval (equation 4, p. 949 of Hall 1988)
The derivation starts from this formula and inherits its validity conditions (i.i.d. sampling, finite moments, etc.). The paper refers readers to Hall (1988) for the assumptions rather than restating them.
standard math Abramowitz and Stegun's asymptotic expansion of Student t quantiles in terms of normal quantiles (equation 7)
Used to replace t_{1-alpha}^{(n-1)} by z_{1-alpha} plus an O(1/n) correction. This is a standard published expansion.

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Cite this review

Pith. "Pith review of Coverage errors for Student's t confidence intervals comparable to those in Hall (1988)." pith.science (2026). https://pith.science/paper/3W4ITQGI

@misc{pith2026250107645,
  author       = {Pith},
  title        = {Pith review of: Coverage errors for Student's t confidence intervals comparable to those in Hall (1988)},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3W4ITQGI}},
  note         = {Machine review of arXiv:2501.07645}
}

abstract

Table 1 of Hall (1988) contains asymptotic coverage error formulas for some nonparametric approximate 95\% confidence intervals for the mean based on $n$ IID samples. The table includes an entry for an interval based on the central limit theorem using Gaussian quantiles and the Gaussian maximum likelihood variance estimate. It is missing an entry for the very widely used Student's $t$ confidence intervals. This note develops such a formula. The impetus to revisit this issue arose from the surprisingly robust performance of confidence intervals based on Student's t statistic in randomized quasi-Monte Carlo sampling. Hall's table had $0.14\kappa -2.12\gamma^2-3.35$ for normal theory intervals; the corresponding entry for Student's $t$ is $0.14\kappa -2.12\gamma^2$. An earlier version of this note reported that it corrected some coverage error formulas in Hall (1988). Two-sided errors take the form $2\Phi^{-1}(0.975)(A\kappa + \gamma^2+C)\varphi(1.96)/n +O(1/n^{3/2})$ where the error may well be $O(n^{-2})$. Hall's table showed $\Phi^{-1}(0.975)(A\kappa + B\gamma^2+C)$. The version intended as a correction had $2(A\kappa + B\gamma^2+C)$, wider by about $2/1.96\doteq1.02$. So, Hall's table really is proportional to the two-sided coverage errors.

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Forward citations

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Works this paper leans on

3 extracted references · 3 canonical work pages · cited by 2 Pith papers

[1]

and Stegun, I

Abramowitz, M. and Stegun, I. A. (1972). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables . National Bureau of Standards, Washington DC. Applied Mathematics Series 55. Tenth Printing

work page 1972
[2]

Hall, P. G. (1988). Theoretical comparisons of bootstrap confidence intervals. The Annals of Statistics , 16(3):927--953

work page 1988
[3]

K., Owen, A

L'Ecuyer, P., Nakayama, M. K., Owen, A. B., and Tuffin, B. (2023). Confidence intervals for randomized quasi-monte carlo estimators. In Corlu, C. G., Hunter, S. R., Lam, H., Onggo, B. S., Shortle, J., and Biller, B., editors, 2023 Winter Simulation Conference (WSC) , pages 445--456. IEEE

work page 2023

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

and Stegun, I

Abramowitz, M. and Stegun, I. A. (1972). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables . National Bureau of Standards, Washington DC. Applied Mathematics Series 55. Tenth Printing

work page 1972

[2] [2]

Hall, P. G. (1988). Theoretical comparisons of bootstrap confidence intervals. The Annals of Statistics , 16(3):927--953

work page 1988

[3] [3]

K., Owen, A

L'Ecuyer, P., Nakayama, M. K., Owen, A. B., and Tuffin, B. (2023). Confidence intervals for randomized quasi-monte carlo estimators. In Corlu, C. G., Hunter, S. R., Lam, H., Onggo, B. S., Shortle, J., and Biller, B., editors, 2023 Winter Simulation Conference (WSC) , pages 445--456. IEEE

work page 2023