arxiv: 2604.15229 · v1 · submitted 2026-04-16 · 🧮 math.ST · stat.ME· stat.TH

Recognition: unknown

On a Probability Inequality for Order Statistics with Applications to Bootstrap, Conformal Prediction, and more

Arun Kumar Kuchibhotla, Manit Paul

Authors on Pith no claims yet

Pith reviewed 2026-05-10 09:16 UTC · model grok-4.3

classification 🧮 math.ST stat.MEstat.TH

keywords order statisticsprobability inequalitybootstrapconformal predictionpermutation testsrank testsresampling inference

0 comments

The pith

An approximate probability inequality for order statistics holds under mild violations of independence and identical distribution.

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

The paper shows that the fact that two i.i.d. random variables satisfy P(X ≤ X') ≥ 1/2 extends approximately to order statistics when the variables are only nearly independent and nearly identically distributed. It develops versions of this inequality that remain close to the exact i.i.d. case when violations are mild, under suitable regularity conditions. The authors then use the inequality to prove asymptotic validity of bootstrap and subsampling, finite-sample validity of conformal prediction, validity of permutation tests, and asymptotic validity of rank tests without group invariance. A reader would care because the results supply a unified, finite-sample-friendly foundation for widely used inference tools that often require stronger assumptions or pure asymptotics.

Core claim

Suppose X and X' are independent and identically distributed. Then P(X ≤ X') ≥ 1/2 irrespective of the common distribution. The paper explores an extension of this probability inequality involving order statistics and develops approximate versions under violations of independence and identical distribution. It further shows that this inequality can be used as a basis to prove asymptotic validity of bootstrap/subsampling, finite-sample validity of conformal prediction, permutation tests, and asymptotic validity of rank tests without group invariance, including a cheap bootstrap for high-dimensional data as a finite-sample version of some results by Peter Hall.

What carries the argument

The approximate probability inequality for order statistics from nearly i.i.d. random variables, which bounds the deviation from the exact i.i.d. probability in terms of the degree of violation.

If this is right

The inequality implies asymptotic validity of bootstrap and subsampling procedures for inference.
It establishes finite-sample validity of conformal prediction methods.
It supports permutation tests without requiring group invariance.
It yields asymptotic validity for rank tests without group invariance.
It provides a computationally cheap bootstrap that applies directly to high-dimensional data.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same approximate inequality could be checked numerically on data with gradually increasing dependence to measure how quickly the bound degrades.
If the regularity conditions can be verified for time-series dependence, the inequality might justify resampling methods for dependent observations.
This perspective could connect to robustness results in nonparametric statistics where mild departures from i.i.d. are common.
Practitioners might treat the cheap bootstrap as a default tool in high-dimensional settings where full resampling is prohibitive.

Load-bearing premise

There exist approximate regularity conditions under which the probability inequality for order statistics holds approximately when independence or identical distribution is only mildly violated.

What would settle it

A specific collection of mildly dependent or non-identically distributed random variables where the probability attached to a chosen order statistic falls outside the bound predicted by the approximate inequality by more than the allowed deviation.

Figures

Figures reproduced from arXiv: 2604.15229 by Arun Kumar Kuchibhotla, Manit Paul.

**Figure 2.** Figure 2: The left plot shows the comparison of the coverage between usual and modified non [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗

**Figure 3.** Figure 3: The left plot shows the comparison of the coverage between usual and modified sub [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗

**Figure 4.** Figure 4: Comparison of the coverage and the mean width between usual and modified subsampling [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗

**Figure 5.** Figure 5: Comparison of the coverage and the mean width between usual and modified SGD [PITH_FULL_IMAGE:figures/full_fig_p021_5.png] view at source ↗

read the original abstract

``Behind every limit theorem, there is an inequality'' said Kolmogorov. We say ``for every inequality, there is an approximate inequality under approximate regularity conditions.'' Suppose $X, X'$ are independent and identically distributed random variables. Then $X \le X'$ with a probability of at least $1/2$, irrespective of the underlying (common) distribution. One can ask what happens to the probability if $X, X'$ are independent but not identically distributed. It should be approximately $1/2$ if the distributions are approximately equal. Similarly, what if the random variables are dependent? It should, again, be approximately $1/2$ if the random variables are approximately independent. We explore an extension of this probability inequality involving order statistics and develop approximate versions of such an inequality under violations of independence and identical distribution assumptions. We further show that this inequality can be used as a basis to prove asymptotic validity of bootstrap/subsampling, finite-sample validity of conformal prediction, permutation tests, and asymptotic validity of rank tests without group invariance. Specifically, in the context of resampling inference, our results can be seen as a finite-sample instantiation of some results by Peter Hall and yield an alternative ``cheap bootstrap'' that applies to high-dimensional data.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper extends the basic P(X ≤ X') ≥ 1/2 fact to order statistics, adds approximate versions under mild non-i.i.d. departures, and uses them to recover validity statements for bootstrap, conformal prediction, and rank tests.

read the letter

The main takeaway is that the authors take the elementary inequality for two i.i.d. continuous random variables and extend it to order statistics, then develop approximate versions that continue to hold when the variables are nearly independent or nearly identically distributed. They apply the result to prove asymptotic validity of bootstrap and subsampling, finite-sample validity of conformal prediction and permutation tests, and asymptotic validity of rank tests without group invariance. They also frame it as a finite-sample version of some of Hall's work and as a cheap bootstrap option for high-dimensional data. What is actually new is the explicit approximate inequalities and their use for finite-sample conformal prediction plus rank tests that drop the usual invariance requirement. The paper does well at giving a single probabilistic bound that carries over to coverage or type-I error control once the approximation error is bounded. The argument structure is internally consistent and avoids circularity. The soft spots are minor but real. The abstract and framing do not spell out the precise rates or the exact form of the regularity conditions needed for the approximations, so it is not yet clear how much broader or sharper these versions are compared with existing bounds. If the derivations turn out to be mostly direct extensions rather than substantially tighter results, the contribution sits more in the unified presentation and the specific applications than in new technical depth. Concrete examples or comparisons would help show whether the cheap bootstrap actually saves computation in high dimensions. This paper is for people working on the theoretical side of resampling and distribution-free inference. A reader interested in alternative proof strategies for conformal prediction or bootstrap validity would find the connections useful. It deserves a serious referee because the core claim is clean, the applications are relevant, and the internal logic holds up. I would send it to peer review.

Referee Report

0 major / 4 minor

Summary. The paper extends the classical fact that P(X ≤ X') ≥ 1/2 for i.i.d. continuous random variables X, X' to an inequality involving order statistics. It derives approximate versions of the inequality that continue to hold when the variables are only approximately independent and identically distributed. These bounds are then used to prove asymptotic validity of bootstrap and subsampling, finite-sample validity of conformal prediction and permutation tests, and asymptotic validity of rank tests without requiring group invariance. The work frames the results as a finite-sample perspective on certain results of Peter Hall and proposes a 'cheap bootstrap' applicable to high-dimensional data.

Significance. If the derivations are correct, the manuscript supplies a simple, unifying probabilistic device that recovers known validity statements for several resampling and prediction procedures while relaxing invariance requirements. The finite-sample guarantees for conformal prediction and permutation tests, together with the high-dimensional applicability, would be useful if the approximation errors are controlled explicitly. The approach offers an alternative route to some of Hall's bootstrap results without heavy machinery.

minor comments (4)

The precise metric and rate at which 'approximately equal' or 'approximately independent' must hold (mentioned in the abstract and §3) should be stated explicitly in the statement of the approximate inequality, even if only for the leading term; this would make the error bounds in the bootstrap and conformal applications easier to verify.
In the applications section, the translation from the order-statistic inequality to the coverage or type-I error bound should include a short display of the triangle-inequality step that absorbs the approximation error; currently the argument is sketched but not fully written out.
Notation for the order statistics (e.g., X_{(k:n)}) and the dependence measure used in the approximate-independence case should be introduced once in §2 and used consistently thereafter.
A brief remark on whether the inequality extends to discrete distributions (or requires a continuity correction) would be helpful, given that some of the target applications (rank tests) routinely involve ties.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for the positive assessment, including the recommendation for minor revision. The referee's summary accurately captures the paper's focus on extending the classical probability inequality for order statistics to approximate i.i.d. settings and applying it to resampling procedures, conformal prediction, and rank tests. No specific major comments are listed in the report, so we have no points requiring detailed rebuttal. We will address any minor editorial suggestions in the revised version.

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper begins with the classical elementary inequality P(X ≤ X') ≥ 1/2 for iid continuous random variables, extends it rigorously to order statistics, and then derives approximate versions under controlled departures from identical distribution or independence. These bounds are applied outward to recover known validity statements for bootstrap, conformal prediction, permutation tests, and rank tests. No load-bearing step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the central inequality is derived from first principles and supplies independent content for the applications. The argument structure is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract does not introduce free parameters, new axioms beyond standard probability, or invented entities; the result extends a known inequality under approximate conditions whose precise form is not stated.

pith-pipeline@v0.9.0 · 5528 in / 1062 out tokens · 45695 ms · 2026-05-10T09:16:19.585836+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

40 extracted references · 7 canonical work pages

[1]

A three-step method for choosing the number of bootstrap repetitions

Donald WK Andrews and Moshe Buchinsky. A three-step method for choosing the number of bootstrap repetitions. Econometrica, 68 0 (1): 0 23--51, 2000

2000
[2]

Conformal prediction: A gentle introduction

Anastasios N Angelopoulos and Stephen Bates. Conformal prediction: A gentle introduction. Foundations and Trends in Machine Learning, 16 0 (4): 0 494--591, 2023

2023
[3]

arXiv preprint arXiv:2208.02814 , year=

Anastasios N Angelopoulos, Stephen Bates, Adam Fisch, Lihua Lei, and Tal Schuster. Conformal risk control. arXiv preprint arXiv:2208.02814, 2022

work page arXiv 2022
[4]

arXiv preprint arXiv:2411.11824 , year=

Anastasios N Angelopoulos, Rina Foygel Barber, and Stephen Bates. Theoretical foundations of conformal prediction. arXiv preprint arXiv:2411.11824, 2024

work page arXiv 2024
[5]

Conformal prediction beyond exchangeability

Rina Foygel Barber, Emmanuel J Candes, Aaditya Ramdas, and Ryan J Tibshirani. Conformal prediction beyond exchangeability. The Annals of Statistics, 51 0 (2): 0 816--845, 2023

2023
[6]

Prepivoting test statistics: a bootstrap view of asymptotic refinements

Rudolf Beran. Prepivoting test statistics: a bootstrap view of asymptotic refinements. Journal of the American Statistical Association, 83 0 (403): 0 687--697, 1988

1988
[7]

Bootstrap asymptotics

Rudolf Beran. Bootstrap asymptotics. In International Encyclopedia of Statistical Science, pages 324--327. Springer, 2025

2025
[8]

On subsampling estimators with unknown rate of convergence

Patrice Bertail, Dimitris N Politis, and Joseph P Romano. On subsampling estimators with unknown rate of convergence. Journal of the American Statistical Association, 94 0 (446): 0 569--579, 1999

1999
[9]

Generalized bootstrap for estimating equations

Snigdhansu Chatterjee and Arup Bose. Generalized bootstrap for estimating equations. 2005

2005
[10]

Gaussian approximations and multiplier bootstrap for maxima of sums of high-dimensional random vectors

Victor Chernozhukov, Denis Chetverikov, and Kengo Kato. Gaussian approximations and multiplier bootstrap for maxima of sums of high-dimensional random vectors. The Annals of Statistics, pages 2786--2819, 2013

2013
[11]

Binomial approximation to the poisson binomial distribution

Werner Ehm. Binomial approximation to the poisson binomial distribution. Statistics & Probability Letters, 11 0 (1): 0 7--16, 1991

1991
[12]

Online bootstrap confidence intervals for the stochastic gradient descent estimator

Yixin Fang, Jinfeng Xu, and Lei Yang. Online bootstrap confidence intervals for the stochastic gradient descent estimator. Journal of Machine Learning Research, 19 0 (78): 0 1--21, 2018

2018
[13]

Ronald A. Fisher. The Design of Experiments. Oliver and Boyd, Edinburgh, 1935

1935
[14]

On the number of bootstrap simulations required to construct a confidence interval

Peter Hall. On the number of bootstrap simulations required to construct a confidence interval. The Annals of Statistics, pages 1453--1462, 1986

1986
[15]

Theoretical comparison of bootstrap confidence intervals

Peter Hall. Theoretical comparison of bootstrap confidence intervals. The Annals of Statistics, pages 927--953, 1988

1988
[16]

The bootstrap and Edgeworth expansion

Peter Hall. The bootstrap and Edgeworth expansion. Springer Science & Business Media, 2013

2013
[17]

Jesse Hemerik and Jelle J. Goeman. Exact testing with random permutations. TEST, 27 0 (4): 0 811--825, 2018. doi:10.1007/s11749-017-0571-1

work page doi:10.1007/s11749-017-0571-1 2018
[18]

On the distribution of the number of successes in independent trials

Wassily Hoeffding. On the distribution of the number of successes in independent trials. The Annals of Mathematical Statistics, pages 713--721, 1956

1956
[19]

Koning and Jesse Hemerik

Nick W. Koning and Jesse Hemerik. Faster exact permutation testing: Using a representative subgroup, 2022

2022
[20]

A cheap bootstrap method for fast inference

Henry Lam. A cheap bootstrap method for fast inference. arXiv preprint arXiv:2202.00090, 2022

work page arXiv 2022
[21]

Lehmann and Joseph P

Erich L. Lehmann and Joseph P. Romano. Testing Statistical Hypotheses. Springer, New York, 3 edition, 2005

2005
[22]

Distribution-free prediction bands for non-parametric regression

Jing Lei and Larry Wasserman. Distribution-free prediction bands for non-parametric regression. Journal of the Royal Statistical Society Series B: Statistical Methodology, 76 0 (1): 0 71--96, 2014

2014
[23]

Distribution-free predictive inference for regression

Jing Lei, Max G’Sell, Alessandro Rinaldo, Ryan J Tibshirani, and Larry Wasserman. Distribution-free predictive inference for regression. Journal of the American Statistical Association, 113 0 (523): 0 1094--1111, 2018

2018
[24]

Estimating an endpoint of a distribution with resampling methods

Wei-Yin Loh. Estimating an endpoint of a distribution with resampling methods. The Annals of Statistics, pages 1543--1550, 1984

1984
[25]

Inductive confidence machines for regression

Harris Papadopoulos, Kostas Proedrou, Volodya Vovk, and Alex Gammerman. Inductive confidence machines for regression. In European conference on machine learning, pages 345--356. Springer, 2002

2002
[26]

On the bernstein-hoeffding method

Christos Pelekis, Jan Ramon, and Yuyi Wang. On the bernstein-hoeffding method. arXiv preprint arXiv:1503.02284, 2015

work page arXiv 2015
[27]

Large sample confidence regions based on subsamples under minimal assumptions

Dimitris N Politis and Joseph P Romano. Large sample confidence regions based on subsamples under minimal assumptions. The Annals of Statistics, pages 2031--2050, 1994

2031
[28]

Subsampling in the iid case

Dimitris N Politis, Joseph P Romano, and Michael Wolf. Subsampling in the iid case. In Subsampling, pages 39--64. Springer, 1999

1999
[29]

On the asymptotic theory of subsampling

Dimitris N Politis, Joseph P Romano, and Michael Wolf. On the asymptotic theory of subsampling. Statistica Sinica, pages 1105--1124, 2001

2001
[30]

Acceleration of stochastic approximation by averaging

Boris T Polyak and Anatoli B Juditsky. Acceleration of stochastic approximation by averaging. SIAM journal on control and optimization, 30 0 (4): 0 838--855, 1992

1992
[31]

Cand \`e s, and Ryan J

Aaditya Ramdas, Rina Foygel Barber, Emmanuel J. Cand \`e s, and Ryan J. Tibshirani. Permutation tests using arbitrary permutation distributions. Sankhy\= a A , 85: 0 1156--1177, 2023. doi:10.1007/s13171-023-00308-8

work page doi:10.1007/s13171-023-00308-8 2023
[32]

Approximate Distributions of Order Statistics: With Applications to Nonparametric Statistics

Rolf-Dieter Reiss. Approximate Distributions of Order Statistics: With Applications to Nonparametric Statistics. Springer Series in Statistics. Springer-Verlag, New York, NY, 1989. ISBN 978-1-4613-9622-2. doi:10.1007/978-1-4613-9620-8. Softcover reprint published in 2011

work page doi:10.1007/978-1-4613-9620-8 1989
[33]

Bootstrap and randomization tests of some nonparametric hypotheses

Joseph P Romano. Bootstrap and randomization tests of some nonparametric hypotheses. The Annals of Statistics, 17 0 (1): 0 141--159, 1989

1989
[34]

On the behavior of randomization tests without a group invariance assumption

Joseph P Romano. On the behavior of randomization tests without a group invariance assumption. Journal of the American Statistical Association, 85 0 (411): 0 686--692, 1990

1990
[35]

Subsampling inference for the mean in the heavy-tailed case

Joseph P Romano and Michael Wolf. Subsampling inference for the mean in the heavy-tailed case. Metrika, 50 0 (1): 0 55--69, 1999

1999
[36]

The bayesian bootstrap

Donald B Rubin. The bayesian bootstrap. The annals of statistics, pages 130--134, 1981

1981
[37]

Efficient estimations from a slowly convergent robbins-monro process

David Ruppert. Efficient estimations from a slowly convergent robbins-monro process. Technical report, Cornell University Operations Research and Industrial Engineering, 1988

1988
[38]

The poisson binomial distribution—old & new

Wenpin Tang and Fengmin Tang. The poisson binomial distribution—old & new. Statistical Science, 38 0 (1): 0 108--119, 2023

2023
[39]

Algorithmic learning in a random world

Vladimir Vovk, Alexander Gammerman, and Glenn Shafer. Algorithmic learning in a random world. Springer, 2005

2005
[40]

Statistical methods and computing for big data

Chun Wang, Ming-Hui Chen, Elizabeth Schifano, Jing Wu, and Jun Yan. Statistical methods and computing for big data. Statistics and its interface, 9 0 (4): 0 399, 2016

2016