One-dimensional quantile-stratified sampling and its application in statistical simulations

Ben O'Neill

REVIEW 3 major objections 6 minor 4 references

One-dimensional quantile-stratified sampling and its application in statistical simulations

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read For a known distribution, drawing one value from each equiprobable quantile block yields sample quantiles that track the true quantiles far more closely than IID sampling.

desk verdict The theory is clean and the extensions are real, but the headline empirical claim about QS importance sampling is riding on a regression with internal reporting errors and diagnosed assumption failures. read the letter →

arxiv 2506.07437 v2 pith:5NKGAKLD submitted 2025-06-09 stat.ME math.STstat.COstat.OTstat.TH

Ben O'Neill This is my paper

classification stat.MEmath.STstat.COstat.OTstat.TH MSC 62D0562G3065C05

keywords quantile-stratifiedsamplingimportanceorderstatisticsLatinhypercubevariancereductionMonteCarlosimulationwithoutreplacementquantilefunction

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

The paper claims that for a known univariate distribution, quantile-stratified (QS) sampling—drawing one value from each of m equiprobable quantile blocks—gives empirical quantiles that adhere to the true quantiles with mean squared error of order O($m^{{-2}}$), versus O($m^{{-1}}$) for IID sampling. Because the QS sample keeps one representative from every block, its order statistics are more regular and its spacings less variable. The paper then inserts QS samples into importance sampling in place of IID samples from the proposal distribution, and reports simulation evidence that the resulting estimator has lower root-mean-squared error, with the relative improvement growing as the sample size increases. If the claim holds, QS importance sampling is a simple, unbiased variance-reduction device that costs no extra samples.

What carries the argument

The central object is the quantile-block partition of the support into m equiprobable intervals [Q((s-1)/m), Q(s/m)], with one value drawn uniformly inside each block after sampling the block indices without replacement. This is the one-dimensional analogue of Latin hypercube sampling. The identities that carry the argument are the moment results for the QS uniform variables (Theorem 1), the distributional results for their order statistics (Theorem 3: U_{(k)}^* ~ U((k-1)/m, k/m), giving MSE 1/($12m^{2}$)), and the spacing distribution (Theorem 4: triangular with variance 1/($6m^{2}$)). These show that the negative correlation produced by without-replacement block selection translates into closer quantile adherence and more regular spacings. For importance sampling, the machinery is the same QS sample drawn from the proposal distribution, with unbiasedness preserved and variance reduced when the transformed composition function G preserves negative correlation.

What would settle it

Simulate QS and IID importance sampling on a distribution pair outside the 20 used (e.g., a multimodal target or a heavy-tailed proposal) at m = 1000 and compare RMSE ratios; if QS does not beat IID, or the ratio does not keep decreasing, the claimed interaction fails. Alternatively, compute the variance of the QS importance-sampling estimator exactly for a smooth G and check whether Var(mu*_m) actually decays like $m^{{-3}}$.

Watch

Extended reading notes

Core claim

The central discovery is that quantile-stratified samples from a known univariate distribution behave like sampling without replacement from the quantile blocks, inducing negative correlation between the underlying uniform values (Theorem 1 gives Cov(U_i^*, U_j^*) = -(m+1)/($12m^{2}$)). This negative correlation makes the order statistics of the QS sample nearly uniform on their blocks (U_{(k)}^* ~ U((k-1)/m, k/m)), giving mean squared error 1/($12m^{2}$) as estimators of p_k^* = (k-1/2)/m and order O($m^{{-2}}$) toward the true quantiles, whereas IID order statistics have MSE of order O($m^{{-1}}$). The same mechanism carries into importance sampling: because the QS sample adheres more closely to the proposal distribution's quantiles, the importance-weighted mean is typically less variable. Simulation over 20 distributions and sample sizes 10 to 1000 is summarized by a regression showing QS estimates with RMSE ratio decreasing from 0.535 at m=10 to 0.093 at m=1000 relative to IID.

Load-bearing premise

The empirical claim that QS importance sampling improves with sample size depends on the regression model in Section 7 being the right specification of log-RMSE over the 20 chosen distributions; the model explains only 30.4% of the variance, and its residuals show heteroskedasticity and a heavy right tail, so the fitted QS-by-log-sample-size interaction could be an artifact of the functional form.

Editorial extensions

If this is right

If the claims hold, any Monte Carlo estimate of a mean for a known univariate distribution can be made more accurate at no extra sample cost by substituting a QS sample for an IID sample.
The improvement grows with sample size, so large-sample simulations benefit most.
Layered QS sampling gives a tunable middle ground: fewer layers approach IID, more layers approach pure QS.
The quasi-QS construction extends the benefit to MCMC settings where only the kernel density is known, without needing the quantile function.
Order-statistic-based uses (like estimating quantiles or constructing QQ plots) inherit the improved adherence.

Reading between the lines

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

If the first-order Taylor approximation in Section 6 is indicative, the QS estimator's variance may scale like m^{-3} for smooth functions, implying a dramatic speedup; a direct variance calculation on a specific smooth G could confirm this.
The regression's low R^2 (30.4%) suggests the 20-distribution design captures only part of the variation; extending to heavier-tailed or multimodal targets could reveal cases where QS underperforms.
The 'improvement increases with sample size' conclusion rests on the QS:log(m) interaction term; since the model explains 30% of variance and shows heteroskedasticity, this interaction may not be universal.
The connection to Latin hypercube sampling suggests a multivariate generalization using randomized orthogonal arrays or Sobol sequences; the paper leaves that extension for future work.

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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

You should know two things about this paper. First, the core theoretical development is solid: quantile-stratified sampling is the one-dimensional case of Latin hypercube sampling, and the paper says so. What's actually new are the layered QS construction (Theorem 2), the quasi-QS results from MCMC output (Theorem 5), and a clear formalization of the order-statistic MSE comparison, O(m^{-2}) vs O(m^{-1}). The appendix proofs check out on a careful read. Second, the empirical section that drives the abstract's headline claim—that QS improvement increases with sample size—is not secure. The Section 7 regression has impossible entries in the coefficient table: the QS main effect has t=1.616 yet p=0.0000 and a confidence interval including 0, and the interaction CI is printed with lower bound greater than upper bound. The model itself explains only 30% of variance, with the paper's own diagnostics acknowledging heteroskedasticity, positive skew, and a heavy right tail. The claim that QS does relatively better at larger m rests entirely on the QS:log(m) interaction, which could easily be an artifact of that misspecification. The two worked examples in Section 6 are nice but don't establish the scaling claim. None of this sinks the theory, because the theory section explicitly allows QS to be worse when the correlation flips. But the empirical generalization needs a direct paired RMSE comparison across sample sizes, with the simulation script and data shipped, before that conclusion is publishable. The paper does mention that the model data and output are available as supplementary files, which is good, but they aren't linked in the text. If they're actually there, the fix is straightforward. My overall read: a solid theoretical contribution to a practical subfield, marred by an overreaching empirical claim. The derivations deserve referee time, and the empirical section is fixable, so I'd send it out rather than desk reject it. Who's it for: people doing univariate Monte Carlo or importance sampling who want a cheap variance-reduction trick. It's not a major reorganization of the field, but the order-statistic results alone are worth a cite. If you're in that area, read the theory and skip to the paired comparison after revision.

Referee Report

3 major / 6 minor

Summary. The paper introduces quantile-stratified (QS) sampling, in which a size- m sample from a known univariate distribution is obtained by partitioning the support into m equiprobable quantile blocks and drawing one value from each block without replacement, in contrast to IID sampling (with replacement). Theorems 1–4 give moment, covariance, and order-statistic results showing that QS uniforms are negatively correlated and that their empirical quantiles have MSE of order O(m^{-2}) versus O(m^{-1}) for IID samples. Theorem 5 extends the analysis to 'quasi-QS' samples generated from MCMC output. A layered generalization (LQS) interpolates between IID and QS sampling. The paper then proposes quantile-stratified importance sampling (QS-IS), gives two worked examples, and reports a simulation study across 20 distributions and five sample sizes, analyzed with a Gaussian regression of log-RMSE on covariates including a QS-by-log-sample-size interaction. The main claims are that QS sampling reduces estimation error relative to IID sampling and that the improvement increases with sample size.

Significance. If the central empirical claim holds, QS-IS is a simple, unbiased variance-reduction device requiring no additional samples, and the order-statistic MSE results are a clean theoretical contribution. The paper's framing of QS as the analogue of SRSWOR for an arbitrary distribution is pedagogically and conceptually useful, and the moment derivations in Theorems 1–5 are laid out in the appendix and appear internally consistent. The machine-checkable nature of the derivations and the public R implementation are commendable. However, the paper's headline conclusion about improvement increasing with sample size rests almost entirely on a single interaction coefficient in a regression whose reported output contains impossible entries and whose assumptions are contradicted by the paper's own diagnostics. The theoretical results establish a clear advantage of QS over IID for estimating quantiles, but they do not by themselves establish the general IS variance-reduction claim; the heuristic in Section 6 explicitly allows QS to be worse when the correlation flips. Thus the significance of the empirical application is currently undermined by the statistical analysis supporting it.

major comments (3)

[Section 7, Coefficient Estimates Table] The reported coefficient table is internally inconsistent: the QS main effect has t = 1.616 yet p = 0.0000 and a 95% confidence interval (-0.05284, 0.54855) that includes 0, while the QS:log-sample-size interaction is reported as -0.37932 with confidence interval (0.44113, 0.31751), whose lower bound exceeds its upper bound. Because the interaction coefficient beta6 = -0.37932 is what drives the claimed improvement-with-sample-size result (multiplicative ratios 0.535 at m=10 down to 0.093 at m=1000), these impossible outputs mean the regression results cannot be used as evidence for that claim. The authors must correct the table, report coherent p-values and confidence intervals, and re-estimate the model before this conclusion can be accepted.
[Section 7, Regression specification and Appendix 2] The model imposes log(sigma_j/sqrt(m)) as an offset, effectively assuming that importance-sampling RMSE scales as sigma_j/sqrt(m). For importance sampling, however, the RMSE also depends on the proposal distribution and the weight function, so this offset need not be an appropriate benchmark; the estimated IID and QS slopes on log(m) are only deviations from a target-SD/sqrt(m) curve that may not describe IS error. Combined with the paper's own diagnostics (heteroskedasticity, positive skew, heavy right tail, R^2 = 0.304), the QS:log(m) interaction could be absorbing misspecification from a few high-RMSE distribution pairs rather than representing a general scaling law. A direct paired comparison of RMSE between QS-IS and IID-IS for each (proposal, target, m) combination is needed to verify the 'improvement increases with sample size' claim. Without such a check, the central empirical conclusion is not secure.
[Section 6, first-order Taylor approximation] The variance approximation leading to Var(mu_hat*) ≈ G'(1/2)^2/(12 m^3) is explicitly a crude first-order result, and the paper acknowledges that QS may be worse when the transformed values have positive correlation. The worked examples in Section 6 show large improvements at m=100, but they are only two instances and cannot support the general claim in the abstract and conclusions that 'the improvement in the resulting estimator increases for higher sample sizes.' The simulation study is the only evidence for the asymptotic trend, and for the reasons above that evidence is currently unreliable. The authors should either strengthen the theoretical argument or present a direct, diagnostic-valid empirical comparison.

minor comments (6)

[Page 2, Introduction] The sentence 'This method has been used in some sampling problems in the fields of hydrology, meteorology and ecology (see e.g., It has been used for other applications (Claggett et al. 2010, ...)' is ungrammatical and contains a missing opening parenthesis or an unfinished phrase; please rephrase.
[Page 25, Example B] The integral for the gamma-distribution example is written with limits 0 to 1, but the gamma density has support on (0, infinity); this appears to be a typo and should be corrected to 0 to infinity.
[Page 22, Section 6] The phrase 'gives a standard deviation that is O(m) larger' is confusing; the intended meaning is that the QS estimator has standard deviation a factor O(m) smaller than the IID estimator (i.e., O(m^{-1}) versus O(m^{-1/2})), and this should be stated clearly.
[Appendix 2] The sentence 'leading to homoskedasticity (higher variance for lower fitted values)' is self-contradictory: non-constant variance is heteroskedasticity, not homoskedasticity. Please correct.
[Page 26, Section 6] The text contains the typo 'important sampling' instead of 'importance sampling' in the sentence 'Improvement of important sampling is just one possible application ...'.
[References and Table 3] The reference to Loh (2008) 'On Latin hypercube sampling' in the Annals of Statistics is cited as volume 24(5), pp. 2058-2080; please verify the volume, year, and page numbers, as these appear inconsistent with standard bibliographic records.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the theoretical results are derived from the QS construction, and the Section 7 conclusions are empirical regression summaries rather than by-construction predictions.

full rationale

The paper's main theoretical claims are self-contained derivations from the sampling construction rather than restatements of its conclusions. Theorem 1 derives the QS uniform moments from SRSWOR sampling theory; Theorems 2-5 derive LQS correlations and order-statistic MSEs from exact distributional facts (uniform, triangular, and beta order-statistic distributions), with external citations only for standard moment formulae. The Section 6 variance comparison is explicitly labeled a crude first-order Taylor approximation, and the paper acknowledges that QS may be worse when the correlation flips, so the heuristic does not smuggle in the conclusion. The broad empirical claim that QS importance sampling improves more at larger sample sizes comes from the Section 7 regression, which is fitted to simulated data and whose estimated multiplicative effects are just re-expressions of the fitted coefficients; this is an empirical summary, not a deduction from the method's definition, and it is not circular even though its robustness is weakened by the appendix's own diagnostics (heteroskedasticity, skew, heavy tails, R-squared 0.304). The single self-citation, O'Neill 2025 for the utilities package's qs.sample function, is only an implementation note in Section 8 and carries no load-bearing argument. No cited uniqueness theorem or ansatz is imported from the author's prior work, and the method is explicitly acknowledged as the one-dimensional case of Latin hypercube sampling with external references. The reported coefficient-table inconsistencies are reporting/correctness concerns, not evidence that the conclusions are equivalent to the inputs by construction.

Assumptions & free parameters 3 free parameters · 8 assumptions · 0 invented entities

The central theory draws on standard sampling and order-statistics mathematics; the empirical claims rest on a hand-chosen simulation design and a fitted regression. The only notable non-standard premise is the IID-order-statistics idealization for the MCMC quasi-QS analysis.

free parameters (3)

Gaussian regression coefficients for LRMSE model = b0=3.78871, b1=0.97403, b3=0.29726, b4=0.08929, b5=0.24786, b6=-0.37932
These are fitted to the 4000 simulated log-RMSE values; the QS-vs-IID conclusion in Section 7 is carried by b5 and especially b6.
MCMC multiplier r = not estimated; user-specified
In quasi-QS sampling, r is an integer chosen by the analyst; Theorem 5 assumes r is large enough for IID-like order statistics.
Layer sizes in illustrative LQS example = m=(18,9,3), K=3
Hand-chosen for Figure 2; not used in the central IS analysis, but affects the reported LQS correlation example.

assumptions (8)

standard math Known finite-population SRSWOR moment formulas (E(S)=(m+1)/2, Var(S)=(m^2-1)/12, Cov=-(m+1)/12).
Used in the proof of Theorem 1 (Appendix 1).
standard math Order statistics of an IID uniform sample have Beta(k, m-k+1) distributions.
Used in Theorems 3 and 4; standard result.
standard math Triangular distribution moment formulas of Kotz and Van Dorpe (2004).
Used in proof of Theorem 4.
domain assumption The target distribution is known, with known and computable quantile function Q.
Stated in Section 1 as the scope of the paper; QS generation depends on Q.
domain assumption The MCMC chain has converged to the stationary distribution and satisfies good coverage diagnostics.
Section 5 assumes burn-in removed and diagnostic criteria met before quasi-QS sampling.
domain assumption For Theorem 5, ordered MCMC values follow standard IID order statistics.
This is the key idealization in the quasi-QS analysis; it equates the MCMC chain's order statistics with IID order statistics.
domain assumption The 20 proposal/target distributions in Section 7 have finite means relevant to the targets and represent a useful test bed.
The simulation conclusions are conditional on this chosen set; the paper does not prove universality.
ad hoc to paper Gaussian linear regression model correctly specifies log-RMSE as a linear function of the chosen covariates.
Used to estimate QS-vs-IID effects; diagnostics show departures.

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Cite this review

Pith. "Pith review of One-dimensional quantile-stratified sampling and its application in statistical simulations." pith.science (2026). https://pith.science/paper/5NKGAKLD

@misc{pith2026250607437,
  author       = {Pith},
  title        = {Pith review of: One-dimensional quantile-stratified sampling and its application in statistical simulations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5NKGAKLD}},
  note         = {Machine review of arXiv:2506.07437}
}

read the original abstract

In this paper we examine quantile-stratified samples from a known univariate probability distribution, with stratification occurring over a partition of the quantile regions in the distribution. We examine some general properties of this sampling method and we contrast it with standard IID sampling to highlight its similarities and differences. We examine the applications of this sampling method to various statistical simulations including importance sampling. We conduct simulation analysis to compare the performance of standard importance sampling against the quantile-stratified importance sampling to see how they each perform on a range of functions.

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Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages

[1]

AND LENG, X-L

BROOKS, S., GELMAN, A., JONES, G. AND LENG, X-L. (EDS) (2011) Handbook of Markov Chain Monte Carlo. Chapman & Hall/CRC: Boca Raton. CLAGGETT, P. R., OKAY, J.A. AND STEHMAN, S. V. (2010) Monitoring regional riparian forest cover change using stratified sampling and multiresolution imagery. Journal of the American Water Resources Association 46(2), pp. 334-...

work page 2011
[2]

■ PROOF OF THEOREM 4: The beta distribution for 𝐷𝐷𝑘𝑘,ℓ is a well-known result for order statistics (see e.g., Reiss 1989, pp

This establishes each of the MSE results and completes the proof. ■ PROOF OF THEOREM 4: The beta distribution for 𝐷𝐷𝑘𝑘,ℓ is a well-known result for order statistics (see e.g., Reiss 1989, pp. 21-22) and so are its moments, so we omit the derivation here. The triangular distribution for 𝐷𝐷𝑘𝑘,ℓ ∗ (see Kotz and Van Dorpe

work page 1989
[113]

(2015) Multivariate ensemble model output statistics using empirical copula

WILKS, D.S. (2015) Multivariate ensemble model output statistics using empirical copula. Quarterly Journal of the Royal Meteorological Society 141(688), pp. 945–952. Page 33 of 43 Appendix 1: Proof of Theorems PROOF OF THEOREM 1: The values 𝑆𝑆1 ∗, … , 𝑆𝑆𝑚𝑚 ∗ are obtained from sampling without replacement from the values 1, … , 𝑚𝑚. As is well-known from sa...

work page 2015
[2004]

Page 38 of 43 Using standard formulae for the moments of this distribution ( Kotz and Van Dorpe 2004, pp

is obtained as: 𝑝𝑝(𝐷𝐷𝑘𝑘,ℓ ∗ = 𝑑𝑑) = 𝑝𝑝(𝑈𝑈(𝑘𝑘+ℓ) ∗ − 𝑈𝑈(𝑘𝑘) ∗ = 𝑑𝑑) = � 𝑝𝑝(𝑈𝑈(𝑘𝑘+ℓ) ∗ = 𝑢𝑢 + 𝑑𝑑) ∙ 𝑝𝑝(𝑈𝑈(𝑘𝑘) ∗ = 𝑢𝑢)𝑑𝑑𝑢𝑢 1 0 = 𝑚𝑚2 � 𝕀𝕀 �𝑘𝑘 + ℓ − 1 𝑚𝑚 < 𝑢𝑢 + 𝑑𝑑 ≤ 𝑘𝑘 + ℓ 𝑚𝑚 � ∙ 𝕀𝕀 �𝑘𝑘 − 1 𝑚𝑚 < 𝑢𝑢 ≤ 𝑘𝑘 𝑚𝑚� 𝑑𝑑𝑢𝑢 1 0 = 𝑚𝑚2 � 𝕀𝕀 �𝑘𝑘 + ℓ − 1 𝑚𝑚 − min � ℓ 𝑚𝑚 , 𝑑𝑑� < 𝑢𝑢 ≤ 𝑘𝑘 𝑚𝑚 − 𝑑𝑑 + min � ℓ 𝑚𝑚 , 𝑑𝑑�� 𝑑𝑑𝑢𝑢 1 0 = ⎩ ⎪⎪⎪ ⎨ ⎪⎪⎪ ⎧ 𝑚𝑚2 � 𝕀𝕀 �𝑘𝑘 + ℓ − 1 𝑚𝑚 − 𝑑𝑑 < 𝑢𝑢 ≤ ...

work page 2004

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

AND LENG, X-L

BROOKS, S., GELMAN, A., JONES, G. AND LENG, X-L. (EDS) (2011) Handbook of Markov Chain Monte Carlo. Chapman & Hall/CRC: Boca Raton. CLAGGETT, P. R., OKAY, J.A. AND STEHMAN, S. V. (2010) Monitoring regional riparian forest cover change using stratified sampling and multiresolution imagery. Journal of the American Water Resources Association 46(2), pp. 334-...

work page 2011

[2] [2]

■ PROOF OF THEOREM 4: The beta distribution for 𝐷𝐷𝑘𝑘,ℓ is a well-known result for order statistics (see e.g., Reiss 1989, pp

This establishes each of the MSE results and completes the proof. ■ PROOF OF THEOREM 4: The beta distribution for 𝐷𝐷𝑘𝑘,ℓ is a well-known result for order statistics (see e.g., Reiss 1989, pp. 21-22) and so are its moments, so we omit the derivation here. The triangular distribution for 𝐷𝐷𝑘𝑘,ℓ ∗ (see Kotz and Van Dorpe

work page 1989

[3] [113]

(2015) Multivariate ensemble model output statistics using empirical copula

WILKS, D.S. (2015) Multivariate ensemble model output statistics using empirical copula. Quarterly Journal of the Royal Meteorological Society 141(688), pp. 945–952. Page 33 of 43 Appendix 1: Proof of Theorems PROOF OF THEOREM 1: The values 𝑆𝑆1 ∗, … , 𝑆𝑆𝑚𝑚 ∗ are obtained from sampling without replacement from the values 1, … , 𝑚𝑚. As is well-known from sa...

work page 2015

[4] [2004]

Page 38 of 43 Using standard formulae for the moments of this distribution ( Kotz and Van Dorpe 2004, pp

is obtained as: 𝑝𝑝(𝐷𝐷𝑘𝑘,ℓ ∗ = 𝑑𝑑) = 𝑝𝑝(𝑈𝑈(𝑘𝑘+ℓ) ∗ − 𝑈𝑈(𝑘𝑘) ∗ = 𝑑𝑑) = � 𝑝𝑝(𝑈𝑈(𝑘𝑘+ℓ) ∗ = 𝑢𝑢 + 𝑑𝑑) ∙ 𝑝𝑝(𝑈𝑈(𝑘𝑘) ∗ = 𝑢𝑢)𝑑𝑑𝑢𝑢 1 0 = 𝑚𝑚2 � 𝕀𝕀 �𝑘𝑘 + ℓ − 1 𝑚𝑚 < 𝑢𝑢 + 𝑑𝑑 ≤ 𝑘𝑘 + ℓ 𝑚𝑚 � ∙ 𝕀𝕀 �𝑘𝑘 − 1 𝑚𝑚 < 𝑢𝑢 ≤ 𝑘𝑘 𝑚𝑚� 𝑑𝑑𝑢𝑢 1 0 = 𝑚𝑚2 � 𝕀𝕀 �𝑘𝑘 + ℓ − 1 𝑚𝑚 − min � ℓ 𝑚𝑚 , 𝑑𝑑� < 𝑢𝑢 ≤ 𝑘𝑘 𝑚𝑚 − 𝑑𝑑 + min � ℓ 𝑚𝑚 , 𝑑𝑑�� 𝑑𝑑𝑢𝑢 1 0 = ⎩ ⎪⎪⎪ ⎨ ⎪⎪⎪ ⎧ 𝑚𝑚2 � 𝕀𝕀 �𝑘𝑘 + ℓ − 1 𝑚𝑚 − 𝑑𝑑 < 𝑢𝑢 ≤ ...

work page 2004