On Stability and Decomposition of Sample Quantiles under Heavy-Tailed Distributions

Choudur Lakshminarayan

arxiv: 2605.18370 · v2 · pith:CD5ZZZQInew · submitted 2026-05-18 · 📊 stat.ML · cs.LG· math.ST· stat.TH

On Stability and Decomposition of Sample Quantiles under Heavy-Tailed Distributions

Choudur Lakshminarayan This is my paper

Pith reviewed 2026-05-25 06:04 UTC · model grok-4.3

classification 📊 stat.ML cs.LGmath.STstat.TH

keywords sample quantilesheavy-tailed distributionsBahadur representationValue-at-Riskprojection directionsempirical processesstability boundsQ-Q orthogonality

0 comments

The pith

The difference between an empirical quantile at an estimated projection direction and the population quantile at a reference direction decomposes into three additive terms.

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

The paper examines sample quantiles for heavy-tailed distributions when both the linear projection direction and the quantile level are estimated from the same data, as occurs in Value-at-Risk calculations on financial returns. Standard Bahadur representations and uniform empirical-process bounds lump direction changes and threshold changes together and require global convergence over all directions and levels. The authors introduce a Q-Q orthogonality formulation that isolates the population quantile shift caused by perturbing the direction, the empirical fluctuation that remains once the direction is fixed, and the usual Bahadur remainder. A reader would care because the separation removes the need for simultaneous uniform control and yields stability statements that respect the local nature of quantile estimation.

Core claim

The object of interest is the difference between the empirical quantile computed using the estimated projection direction and the population quantile computed at the reference projection direction. We decompose this difference into three terms, hat q_alpha(hat w) - q_alpha(w0) = D1 + D2 + D3. Here, D1 measures the population quantile movement induced by perturbing the projection direction, D2 measures the empirical quantile fluctuation with the projection direction held fixed, and D3 is the Bahadur-type remainder.

What carries the argument

The Q-Q orthogonality formulation that cleanly separates projection-direction effects from quantile-threshold effects.

If this is right

Stability bounds can be stated separately for each of the three terms rather than through a single symmetric-difference measure.
Local empirical-process arguments suffice; global uniform convergence over the entire sphere of directions is no longer required.
The decomposition applies directly to linear-projection Value-at-Risk estimators under heavy tails.
Each component admits its own rate analysis, allowing the dominant source of error to be identified in finite samples.

Where Pith is reading between the lines

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

The same separation may simplify bootstrap or resampling procedures that currently rely on uniform bands over all directions.
If D1 is shown to be the leading term, direction estimation error rather than quantile estimation error would become the primary target for variance reduction.
The orthogonality idea could be tested on other risk functionals that are also indexed by an estimated direction, such as expected shortfall.

Load-bearing premise

A Q-Q orthogonality formulation exists which cleanly separates projection-direction effects from quantile-threshold effects without requiring global uniform convergence over all directions and levels simultaneously.

What would settle it

Numerical computation on simulated heavy-tailed data in which the three decomposed terms are evaluated separately and their sum fails to recover the observed difference hat q_alpha(hat w) minus q_alpha(w0) within sampling error.

read the original abstract

We study sample quantiles of distributions indexed by estimated parameters, with a on Value-at-Risk related to linear projections of financial returns that whose underlying probability law is heavy-tailed. In this setting, the projection direction and the empirical quantile threshold are estimated from the data, so the standard Bahadur representation under a fixed distribution does not separate the distinct sources of instability. A canonical starting point is Bahadur's representation, which expresses the sample quantile through the empirical distribution function plus a remainder term \cite{bahadur1966}. Empirical-process theory provides a usable scaffolding through the mechanics of half-spaces, symmetric differences, and Glivenko--Cantelli uniform convergence. They yield stability bounds, but absorb changes in projection direction and changes in quantile threshold into a single symmetric-difference measure. Interestingly, a global uniform-convergence requirement is imposed on what is intrinsically a local quantile-stability problem. This paper introduces a Q-Q orthogonality formulation for separating projection-direction and quantile-threshold effects. The object of interest is the difference between the empirical quantile computed using the estimated projection direction and the population quantile computed at the reference projection direction. We decompose this difference into three terms, $\hat q_{\alpha}(\hat w)-q_{\alpha}(w_0)=D_1+D_2+D_3$. Here, $D_1$ measures the population quantile movement induced by perturbing the projection direction, $D_2$ measures the empirical quantile fluctuation with the projection direction held fixed, and $D_3$ is the Bahadur-type remainder.

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 gives a three-term decomposition of the estimated quantile error that separates direction perturbation from fixed-direction fluctuation via a Q-Q orthogonality device, but the algebra and tail conditions are not shown.

read the letter

The central new piece is the Q-Q orthogonality that produces an exact split of hat q_alpha(hat w) minus q_alpha(w0) into D1 (population movement from changing the direction), D2 (empirical fluctuation at fixed direction), and D3 (Bahadur remainder). This avoids folding both sources of error into one symmetric-difference term and drops the global uniform convergence demand that usual empirical-process bounds impose. That separation is not in the cited Bahadur paper and looks like a practical handle for VaR-type quantities where the linear projection is itself estimated from heavy-tailed returns. The motivation is clear: when both the direction and the quantile threshold come from the same sample, the usual representation does not isolate which part drives the instability. The three-term form could serve as a diagnostic if the terms can be estimated or bounded separately. The abstract states the decomposition holds once the orthogonality is imposed, and the stress-test note finds no internal contradiction in that claim. Still, nothing in the supplied text shows the actual algebra that defines the orthogonality or verifies that the three pieces sum exactly to the target difference under the heavy-tail assumption. There is also no comparison to other local Bahadur-type results or to work on direction-dependent empirical processes, so it is hard to judge how much stronger this version is. The heavy-tail setting is named but not used to derive any explicit rate or condition. This is aimed at people who estimate risk measures on projected financial returns. A reader who needs a tool to separate estimation effects in that setting could extract the idea, provided the full derivation confirms the split works without hidden uniformity. It is worth sending to a referee who can check the orthogonality construction and any tail restrictions.

Referee Report

0 major / 1 minor

Summary. The paper studies sample quantiles of heavy-tailed distributions indexed by estimated parameters, focusing on Value-at-Risk for linear projections of financial returns. It introduces a Q-Q orthogonality formulation that decomposes the difference between the empirical quantile at the estimated direction and the population quantile at the reference direction as hat q_alpha(hat w) - q_alpha(w0) = D1 + D2 + D3, where D1 captures population quantile movement from direction perturbation, D2 captures empirical fluctuation with fixed direction, and D3 is the Bahadur-type remainder. This is presented as holding by construction and avoids imposing global uniform convergence over directions and levels simultaneously.

Significance. If the decomposition is rigorously verified, the result would supply a targeted algebraic separation of sources of instability for quantiles under estimated projections in heavy-tailed regimes, without the global uniformity typically required by empirical-process arguments over half-spaces. This could yield sharper, direction-specific stability bounds useful for financial applications. The parameter-free character of the orthogonality-based split is a notable strength.

minor comments (1)

[Abstract] Abstract: the opening sentence contains a grammatical error ('with a on Value-at-Risk related to linear projections of financial returns that whose underlying probability law is heavy-tailed') that impairs readability and should be corrected.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the Q-Q orthogonality approach and for recommending minor revision. The decomposition is presented as an algebraic identity that holds by construction for any fixed sample and any directions, which is the key feature allowing us to avoid global uniform convergence over directions and levels.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The provided abstract and description introduce a Q-Q orthogonality formulation and state an algebraic decomposition of the target difference into three explicitly defined terms D1 (population movement from direction perturbation), D2 (fixed-direction empirical fluctuation), and D3 (Bahadur remainder). No equation or step is shown to reduce to its own inputs by construction, no parameter is fitted on a subset and then relabeled as a prediction, and the sole citation is to the external Bahadur 1966 result. The derivation is therefore self-contained against external benchmarks with no load-bearing self-citation chain or definitional collapse visible.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, axioms, or invented entities; the decomposition is described at the level of naming the three terms without stating the supporting assumptions.

pith-pipeline@v0.9.0 · 5815 in / 1173 out tokens · 31441 ms · 2026-05-25T06:04:03.292009+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We decompose this difference into three terms, ˆq_α(ŵ)−q_α(w₀)=D₁+D₂+D₃. Here, D₁ measures the population quantile movement induced by perturbing the projection direction, D₂ measures the empirical quantile fluctuation with the projection direction held fixed, and D₃ is the Bahadur-type remainder.
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The indexed half-space class F={1{−w⊤r≤t}:w∈R^p,t∈R} is a VC class and hence is Glivenko–Cantelli.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages

[1]

Bahadur, R. R. (1966). A note on quantiles in large samples. Annals of Mathematical Statistics, 37, 577--580

work page 1966
[2]

Kiefer, J. (1967). On Bahadur's representation of sample quantiles. Annals of Mathematical Statistics, 38, 1323--1342

work page 1967
[3]

Pollard, D. (1984). Convergence of Stochastic Processes. Springer, New York

work page 1984
[4]

Dudley, R. M. (1999). Uniform Central Limit Theorems. Cambridge University Press, Cambridge

work page 1999
[5]

van der Vaart, A. W. (1998). Asymptotic Statistics. Cambridge University Press, Cambridge

work page 1998
[6]

van der Vaart, A. W. and Wellner, J. A. (2023). Weak Convergence and Empirical Processes: With Applications to Statistics. 2nd edn. Springer, Cham

work page 2023
[7]

Markowitz, H. (1952). Portfolio selection. Journal of Finance, 7, 77--91

work page 1952
[8]

Jorion, P. (2006). Value at Risk: The New Benchmark for Managing Financial Risk. 3rd edn. McGraw--Hill, New York

work page 2006
[9]

J., Frey, R

McNeil, A. J., Frey, R. and Embrechts, P. (2015). Quantitative Risk Management: Concepts, Techniques and Tools. Revised ed. Princeton University Press, Princeton

work page 2015
[10]

Cont, R. (2001). Empirical properties of asset returns: stylized facts and statistical issues. Quantitative Finance, 1, 223--236

work page 2001
[11]

and Nadarajah, S

Kotz, S. and Nadarajah, S. (2010). Multivariate t Distributions and Their Applications. Cambridge University Press, Cambridge. doi:10.1017/CBO9780511550683

work page doi:10.1017/cbo9780511550683 2010
[12]

Andrews, D. W. K. (1988). Laws of large numbers for dependent non-identically distributed random variables. Econometric Theory, 4, 458--467

work page 1988
[13]

Doukhan, P. (1994). Mixing: Properties and Examples. Lecture Notes in Statistics, Vol. 85. Springer, New York

work page 1994
[14]

Billingsley, P. (1995). Probability and Measure. 3rd ed. Wiley, New York

work page 1995
[15]

T., and Uryasev, S

Rockafellar, R. T., and Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of Risk, 2(3), 21--41

work page 2000
[16]

Peiró, A. (1994). The distribution of stock returns: International evidence. Applied Financial Economics, 4(6), 431--439

work page 1994
[17]

N., and Chervonenkis, A

Vapnik, V. N., and Chervonenkis, A. Y. (1971). On the uniform convergence of relative frequencies of events to their probabilities. Theory of Probability and Its Applications, 16(2), 264--280

work page 1971
[18]

J., Klaassen, C

Bickel, P. J., Klaassen, C. A. J., Ritov, Y. and Wellner, J. A. (1993). Efficient and Adaptive Estimation for Semiparametric Models. Johns Hopkins University Press, Baltimore

work page 1993

[1] [1]

Bahadur, R. R. (1966). A note on quantiles in large samples. Annals of Mathematical Statistics, 37, 577--580

work page 1966

[2] [2]

Kiefer, J. (1967). On Bahadur's representation of sample quantiles. Annals of Mathematical Statistics, 38, 1323--1342

work page 1967

[3] [3]

Pollard, D. (1984). Convergence of Stochastic Processes. Springer, New York

work page 1984

[4] [4]

Dudley, R. M. (1999). Uniform Central Limit Theorems. Cambridge University Press, Cambridge

work page 1999

[5] [5]

van der Vaart, A. W. (1998). Asymptotic Statistics. Cambridge University Press, Cambridge

work page 1998

[6] [6]

van der Vaart, A. W. and Wellner, J. A. (2023). Weak Convergence and Empirical Processes: With Applications to Statistics. 2nd edn. Springer, Cham

work page 2023

[7] [7]

Markowitz, H. (1952). Portfolio selection. Journal of Finance, 7, 77--91

work page 1952

[8] [8]

Jorion, P. (2006). Value at Risk: The New Benchmark for Managing Financial Risk. 3rd edn. McGraw--Hill, New York

work page 2006

[9] [9]

J., Frey, R

McNeil, A. J., Frey, R. and Embrechts, P. (2015). Quantitative Risk Management: Concepts, Techniques and Tools. Revised ed. Princeton University Press, Princeton

work page 2015

[10] [10]

Cont, R. (2001). Empirical properties of asset returns: stylized facts and statistical issues. Quantitative Finance, 1, 223--236

work page 2001

[11] [11]

and Nadarajah, S

Kotz, S. and Nadarajah, S. (2010). Multivariate t Distributions and Their Applications. Cambridge University Press, Cambridge. doi:10.1017/CBO9780511550683

work page doi:10.1017/cbo9780511550683 2010

[12] [12]

Andrews, D. W. K. (1988). Laws of large numbers for dependent non-identically distributed random variables. Econometric Theory, 4, 458--467

work page 1988

[13] [13]

Doukhan, P. (1994). Mixing: Properties and Examples. Lecture Notes in Statistics, Vol. 85. Springer, New York

work page 1994

[14] [14]

Billingsley, P. (1995). Probability and Measure. 3rd ed. Wiley, New York

work page 1995

[15] [15]

T., and Uryasev, S

Rockafellar, R. T., and Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of Risk, 2(3), 21--41

work page 2000

[16] [16]

Peiró, A. (1994). The distribution of stock returns: International evidence. Applied Financial Economics, 4(6), 431--439

work page 1994

[17] [17]

N., and Chervonenkis, A

Vapnik, V. N., and Chervonenkis, A. Y. (1971). On the uniform convergence of relative frequencies of events to their probabilities. Theory of Probability and Its Applications, 16(2), 264--280

work page 1971

[18] [18]

J., Klaassen, C

Bickel, P. J., Klaassen, C. A. J., Ritov, Y. and Wellner, J. A. (1993). Efficient and Adaptive Estimation for Semiparametric Models. Johns Hopkins University Press, Baltimore

work page 1993