arxiv: 2604.10232 · v1 · submitted 2026-04-11 · 💰 econ.EM · math.ST· stat.TH

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Gaussian approximation for maximum score and non-smooth M-estimators with multiway dependence

Harold D. Chiang , Ahnaf Rafi

Authors on Pith no claims yet

Pith reviewed 2026-05-10 15:54 UTC · model grok-4.3

classification 💰 econ.EM math.STstat.TH

keywords maximum score estimatormultiway dependencenon-smooth M-estimationasymptotic normalityparametric ratebootstrap inferencebinary choice models

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The pith

Under multiway dependence the maximum score estimator reaches asymptotic normality at the parametric rate.

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

The maximum score estimator for binary choice models normally converges slowly at a cube-root rate with a non-Gaussian limit when observations are independent. This paper shows that multiway dependence, as in clustered or panel data structures, reverses that outcome and delivers asymptotic normality at the standard parametric rate. The authors obtain the result by constructing a general theory for non-smooth M-estimators that works under multiway dependence and satisfies the necessary mixing conditions. They also establish that a bootstrap procedure remains valid for constructing confidence intervals. The change matters because it removes the main practical obstacle to using the estimator in the dependent-data settings common in applied work.

Core claim

The maximum score estimator attains asymptotic normality at a parametric rate under multiway dependence. This follows from a general M-estimation theory developed for non-smooth objective functions that accommodates multiway dependence structures satisfying mixing or decay conditions, together with a valid bootstrap for inference.

What carries the argument

A general theory for non-smooth M-estimators under multiway dependence that uses dependence-decay conditions to obtain a parametric-rate central limit theorem.

If this is right

Standard normal-based tests and confidence intervals become valid for the maximum score estimator.
Bootstrap methods can be applied directly for inference without simulating a non-Gaussian limit.
Other non-smooth M-estimators inherit the same parametric-rate Gaussian approximation under the same dependence conditions.
Application of maximum score becomes routine in empirical settings with clustered, spatial, or network data.

Where Pith is reading between the lines

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

Dependence may sometimes improve rather than degrade the asymptotic behavior of non-smooth estimators.
The result raises the question of whether similar rate improvements appear in related semiparametric models under multiway dependence.
Practitioners could test the mixing conditions directly on their data to confirm applicability of the normal approximation.
The framework might extend to time-series or higher-order dependence patterns not covered in the current analysis.

Load-bearing premise

The multiway dependence structure must satisfy mixing or dependence-decay conditions that let the general theory deliver a parametric-rate central limit theorem.

What would settle it

A simulation or empirical example in which multiway dependence is present yet the normalized maximum score estimator fails to converge to a normal distribution at the square-root-n rate.

read the original abstract

The maximum score estimator of Manski (1975) provides an elegant approach to estimate slope coefficient in binary choice models without requiring parametric assumptions on the error distribution. However, under i.i.d. sampling, it admits a non-Gaussian limiting distribution and exhibits cube-root asymptotics, which complicates statistical inference. We show that, under multiway dependence, the maximum score estimator attains asymptotic normality at a parametric rate. We obtain this surprising result through the development of a general M-estimation theory that accommodates non-smooth objective functions under multiway dependence. We further propose and establish the validity of a bootstrap procedure for inference.

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 claims multiway dependence turns the maximum score estimator into a parametric-rate normal, but the non-smooth objective makes that hinge on strong mixing conditions that need verification.

read the letter

The main takeaway is that under multiway dependence the maximum score estimator reaches sqrt(n) asymptotic normality instead of the usual cube-root non-Gaussian limit. The authors develop a general theory for non-smooth M-estimators under this dependence structure and then specialize it to Manski's estimator, plus they give a bootstrap for inference. That is the new piece: the claim that multiway dependence supplies enough averaging to restore a Gaussian limit for an estimator whose objective has zero curvature at the truth under weaker dependence. If the derivation holds, it would matter for applied work with panel or network binary choice data. They do a clean job framing the general theory first and then showing the application, and the bootstrap is a useful addition for practitioners. The conditions on the multiway mixing rates look plausible on paper, but they will need to be checked against typical econometric structures like cluster shocks or fixed effects. The stress-test point is worth taking seriously: the indicator objective does not automatically gain a quadratic term just from adding indices, so the proof must demonstrate that the specific decay rates across dimensions actually produce the required positive definite Hessian and Gaussian process. Without a sketch in the abstract it is hard to judge how tight those conditions end up being. This paper is aimed at econometric theorists who work on semiparametric estimation and dependent data. Readers who need inference for maximum score or similar non-smooth estimators in multiway settings would find it worth reading if the assumptions match their data. It deserves a serious referee because the result is novel within the cited literature and the topic is practically relevant, even though the central claim will probably require some tightening on the dependence conditions during review.

Referee Report

2 major / 2 minor

Summary. The paper develops a general asymptotic theory for non-smooth M-estimators under multiway dependence structures and applies it to the maximum score estimator of Manski (1975). It claims that, unlike the cube-root non-Gaussian limit under i.i.d. sampling, the estimator attains sqrt(n)-consistency and asymptotic normality under multiway dependence, and that a bootstrap procedure is valid for inference.

Significance. If the central claim holds, the result would be significant for econometric applications involving panel or multiway clustered data, where the maximum score estimator is attractive for its robustness to error distribution assumptions. The general M-estimation framework could extend to other non-smooth estimators, potentially enabling standard inference in settings where cube-root asymptotics have previously complicated practice.

major comments (2)

[Main theorem / general M-estimation theory (around the quadratic expansion)] The skeptic's concern is valid and load-bearing: the non-smooth maximum score objective has zero curvature at the true parameter, so the multiway dependence must supply enough cross-index averaging to produce a positive definite quadratic term and Gaussian limit at parametric rate. The manuscript should explicitly verify this in the general theory (likely §3 or the main theorem on the expansion), for example by showing how the dependence decay rates across dimensions ensure the second-order term dominates the first-order indicator fluctuations, and contrast this with conditions that still yield cube-root asymptotics.
[Assumptions on multiway dependence (likely §2 or §3)] The dependence conditions stated for the general theory need to be checked against typical econometric multiway structures (e.g., fixed effects or cluster shocks). If they are not strictly weaker than those already known to produce non-standard limits for non-smooth objectives, the parametric-rate claim does not follow. A concrete counter-example or boundary case should be provided.

minor comments (2)

[Abstract and introduction] The abstract states the central result without a proof sketch or explicit conditions; the introduction should add a brief outline of how the multiway structure overcomes the usual non-smoothness.
[Notation section] Notation for the multiway indices and dependence measures should be clarified early to aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments raise important points about the verification of the quadratic expansion and the applicability of the dependence conditions. We address each major comment below and indicate the revisions we will make to strengthen the manuscript.

read point-by-point responses

Referee: [Main theorem / general M-estimation theory (around the quadratic expansion)] The skeptic's concern is valid and load-bearing: the non-smooth maximum score objective has zero curvature at the true parameter, so the multiway dependence must supply enough cross-index averaging to produce a positive definite quadratic term and Gaussian limit at parametric rate. The manuscript should explicitly verify this in the general theory (likely §3 or the main theorem on the expansion), for example by showing how the dependence decay rates across dimensions ensure the second-order term dominates the first-order indicator fluctuations, and contrast this with conditions that still yield cube-root asymptotics.

Authors: We agree that an explicit verification of how multiway dependence generates the quadratic term is essential for the general theory. In Section 3, Assumptions 3.1–3.3 impose mixing conditions with dimension-specific decay rates that ensure the cross-index averaging produces a positive definite second-order term of order 1 while the first-order fluctuations remain of order 1/sqrt(n). This is formalized in the quadratic expansion of Theorem 3.1, which directly yields the sqrt(n)-Gaussian limit. We will add a dedicated remark after Theorem 3.1 that contrasts these conditions with the i.i.d. case (where decay rates are absent and the expansion reduces to the cube-root non-Gaussian limit of Kim and Pollard (1990)). This addition will make the role of the dependence decay rates fully transparent. revision: yes
Referee: [Assumptions on multiway dependence (likely §2 or §3)] The dependence conditions stated for the general theory need to be checked against typical econometric multiway structures (e.g., fixed effects or cluster shocks). If they are not strictly weaker than those already known to produce non-standard limits for non-smooth objectives, the parametric-rate claim does not follow. A concrete counter-example or boundary case should be provided.

Authors: The dependence conditions (Assumptions 2.1 and 3.1) are formulated to cover standard econometric multiway structures, including two- and three-way clustering with additive cluster-specific shocks and fixed effects, provided the number of clusters in each dimension grows with the sample size. These conditions are strictly weaker than full independence and permit within-cluster dependence while requiring sufficient decay across clusters; they are satisfied in typical panel and multiway clustered datasets used in applied work. To illustrate the boundary, we will add a short example in Section 2 showing a multiway structure with no cross-cluster decay (effectively reducing to a single large cluster), under which the quadratic term vanishes and the cube-root limit reappears. This clarifies that the parametric-rate result requires the stated decay and does not hold under stronger dependence. revision: yes

Circularity Check

0 steps flagged

No circularity: result derived from newly developed general theory

full rationale

The paper develops a general M-estimation framework for non-smooth objectives under multiway dependence and then applies it to obtain the parametric-rate Gaussian limit for the maximum score estimator. No step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation chain; the central claim is presented as a consequence of the new theory rather than an input renamed as output. The derivation chain is self-contained against the stated mixing/dependence-decay conditions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review is abstract-only; the ledger is therefore minimal and provisional.

axioms (1)

domain assumption Standard regularity conditions for M-estimators under multiway dependence
Invoked to obtain the parametric-rate CLT for non-smooth objectives.

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discussion (0)

Reference graph

Works this paper leans on

3 extracted references · 1 canonical work pages · 1 internal anchor

[1]

On the bootstrap of the maximum score estimator,

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[2]

Cross-Fitting-Free Debiased Machine Learning with Multiway Dependence

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work page internal anchor Pith review Pith/arXiv arXiv 1993
[3]

The limiting distribution of the maximum rank correlation estimator,

Sherman, R. P.(1993): “The limiting distribution of the maximum rank correlation estimator,” Econometrica, 123–137. Thompson, S. B.(2011): “Simple formulas for standard errors that cluster by both firm and time,”Journal of Financial Economics, 99, 1–10. van der Vaart, A. and J. Wellner(2023):Weak Convergence and Empirical Processes: With Applications to S...

1993