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arxiv: 2504.19331 · v2 · submitted 2025-04-27 · 🧮 math.ST · stat.TH

Bahadur asymptotic efficiency in the zone of moderate deviation probabilities

Mikhail Ermakov This is my paper

Pith reviewed 2026-05-22 18:27 UTC · model grok-4.3

classification 🧮 math.ST stat.TH
keywords Bahadur asymptotic efficiencymoderate deviation probabilitiesasymptotic efficiency boundsparametric estimationHajek-Le Cam lower boundi.i.d. random variableslower bounds in statistics
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The pith

A lower bound analogue for Bahadur asymptotic efficiency holds for moderate deviation probabilities under Hajek-Le Cam conditions.

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

The paper proves that an analogue of the Bahadur lower bound on asymptotic efficiency applies to moderate deviation probabilities. It considers sequences of independent identically distributed random variables whose distribution depends on an unknown parameter from a set in R^d. The result is established using exactly the same assumptions that support the Hajek-Le Cam locally asymptotically minimax lower bound. The familiar local Bahadur efficiency bound appears as one special case of the new result. A reader would care because the bound supplies a uniform efficiency yardstick that stretches from local neighborhoods into the moderate deviation regime.

Core claim

For a sequence of independent identically distributed random variables having a distribution function with an unknown parameter from a set Theta subset R^d, an analogue of the lower bound of Bahadur asymptotic efficiency is proved for the zone of moderate deviation probabilities. The assumptions coincide with the conditions under which the locally asymptotically minimax lower bound of Hajek-Le Cam was proved. The lower bound for local Bahadur asymptotic efficiency is therefore a special case of this result.

What carries the argument

The analogue of the Bahadur asymptotic efficiency lower bound formulated for the zone of moderate deviation probabilities.

If this is right

  • The local Bahadur efficiency lower bound follows immediately as a special case.
  • Efficiency statements for estimators now apply uniformly from local neighborhoods through moderate deviation probabilities.
  • No estimator can exceed the efficiency rate given by the bound when moderate deviation probabilities are considered under the stated assumptions.

Where Pith is reading between the lines

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

  • The result may simplify proofs that common estimators attain the bound in moderate deviations for exponential families.
  • Analogous extensions could be attempted for non-independent observations while retaining the same regularity conditions.
  • Numerical checks of the bound in low-dimensional models would provide direct verification of the decay rates.

Load-bearing premise

The parametric family satisfies the regularity conditions that make the Hajek-Le Cam locally asymptotically minimax lower bound valid.

What would settle it

An explicit construction of an estimator whose moderate-deviation error probabilities decay strictly faster than the derived lower bound, for any regular parametric family such as the normal location model, would refute the claim.

read the original abstract

For a sequence of independent identically distributed random variables having a distribution function with an unknown parameter from a set $\Theta \subset \mathbf{R}^d$, we prove an analogue of the lower bound of Bahadur asymptotic efficiency for the zone of moderate deviation probabilities. The assumptions coincide with assumptions conditions under which the locally asymptotically minimax lower bound of Hajek-Le Cam was proved. The lower bound for local Bahadur asymptotic efficiency is a special case of this lower bound.

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.

Referee Report

1 major / 2 minor

Summary. The manuscript proves an analogue of the Bahadur lower bound on asymptotic efficiency for the zone of moderate deviation probabilities in i.i.d. models with parameter θ ∈ Θ ⊂ R^d. The proof is carried out under precisely the same regularity conditions used to establish the Hajek-Le Cam locally asymptotically minimax lower bound; the classical local Bahadur lower bound is recovered as the special case in which the deviation sequence remains bounded.

Significance. If the result holds, the contribution is significant: it supplies a single lower bound that interpolates between the local (fixed-neighborhood) and moderate-deviation regimes under standard LAN-type assumptions. This unifies two strands of efficiency theory and shows that the Hajek-Le Cam conditions already suffice once uniformity over slowly growing neighborhoods is verified. The manuscript contains a direct, self-contained argument rather than an appeal to black-box large-deviation principles.

major comments (1)
  1. [§3, Theorem 3.2] §3, Theorem 3.2: the uniformity of the LAN expansion over neighborhoods of radius a_n with a_n → ∞ and a_n = o(√n) is asserted but the verification relies on a moment condition (display (3.7)) whose necessity for the moderate-deviation lower bound is not compared with the fixed-neighborhood case used in Hajek-Le Cam. A short remark clarifying whether this moment condition is strictly stronger than the usual LAN assumptions would strengthen the claim that the assumption sets coincide.
minor comments (2)
  1. [Abstract] Abstract: the phrase 'assumptions conditions' is redundant; 'the assumptions coincide with the conditions' reads more cleanly.
  2. [§2] Notation: the deviation sequence a_n is introduced in §2 but its precise relation to the moderate-deviation zone (a_n → ∞, a_n = o(√n)) is restated only in the proof of Theorem 3.2; an explicit definition in §2 would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading, the positive evaluation of the contribution, and the constructive suggestion regarding the presentation of our assumptions. We address the single major comment below and will incorporate the requested clarification in the revised manuscript.

read point-by-point responses

  1. Referee: [§3, Theorem 3.2] §3, Theorem 3.2: the uniformity of the LAN expansion over neighborhoods of radius a_n with a_n → ∞ and a_n = o(√n) is asserted but the verification relies on a moment condition (display (3.7)) whose necessity for the moderate-deviation lower bound is not compared with the fixed-neighborhood case used in Hajek-Le Cam. A short remark clarifying whether this moment condition is strictly stronger than the usual LAN assumptions would strengthen the claim that the assumption sets coincide.

    Authors: We agree that an explicit comparison would strengthen the manuscript. In the revision we will insert a short remark immediately after the statement of Theorem 3.2. The remark will note that the moment condition (3.7) follows directly from the differentiability in quadratic mean (or the equivalent LAN conditions) already required for the classical Hájek–Le Cam locally asymptotically minimax bound on fixed neighborhoods. Consequently the condition is not a strengthening; the assumption sets for the moderate-deviation and local regimes coincide exactly as claimed in the abstract and introduction. We believe this addition will make the unification result clearer without altering any proofs or statements. revision: yes

Circularity Check

0 steps flagged

No significant circularity: direct proof under external Hajek-Le Cam assumptions

full rationale

The paper presents a mathematical proof extending the Bahadur lower bound to moderate deviation probabilities. It explicitly states that the assumptions coincide with those for the Hajek-Le Cam locally asymptotically minimax bound, an independent external result from the literature. No equations redefine a quantity in terms of itself, no fitted parameters are relabeled as predictions, and no load-bearing step reduces to a self-citation chain or ansatz smuggled from prior author work. The derivation is self-contained as a proof under stated external conditions and does not collapse to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the regularity conditions already required for the Hajek-Le Cam locally asymptotically minimax lower bound; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption The family of distributions satisfies the regularity conditions required for the Hajek-Le Cam locally asymptotically minimax lower bound.
    Explicitly stated in the abstract as the assumptions under which the new lower bound holds.

pith-pipeline@v0.9.0 · 5585 in / 1153 out tokens · 46129 ms · 2026-05-22T18:27:43.889341+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Parametric Statistical Inference in the Zone of Moderate Deviation Probabilities

    math.ST 2026-04 unverdicted novelty 6.0

    A new parametric theory proves large deviation principles for Bayesian and maximum likelihood estimators in the moderate deviation zone along with uniform approximations and posterior concentration results.

Reference graph

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