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arxiv: 2605.02112 · v1 · submitted 2026-05-04 · 📊 stat.ME

An adaptive variance estimator for relative sparsity

Samuel Julian Weisenthal This is my paper

Pith reviewed 2026-05-08 19:33 UTC · model grok-4.3

classification 📊 stat.ME
keywords theoremvarianceadaptiveestimatorfullypolicyrelativeselection
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The pith

A new adaptive variance estimator for relative sparsity coefficients is introduced that fully utilizes the prior asymptotic normality theorem and incorporates variable selection effects.

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

Relative sparsity is a statistical approach used in policy learning where only a subset of available variables are selected as important for making decisions, such as treatment policies in medicine. Earlier work created methods for inference under relative sparsity and proved an asymptotic normality result for the adaptive lasso estimator, but that result was not completely applied when calculating the variance of the selected policy coefficients. This paper creates a new variance estimator designed to use the full theorem and to adjust for the fact that variable selection has already occurred. The goal is to produce more accurate uncertainty measures that can be shown in graphical selection diagrams. These diagrams help visualize which variables were chosen and how reliable the choices are. The authors argue this will support safer use of such methods when learning policies for clinical practice.

Core claim

Here, we develop a new coefficient variance estimator that fully uses this theorem and, in the process, takes into account the variable selection.

Load-bearing premise

That the adaptive lasso asymptotic normality theorem from prior work applies directly to the new variance estimator and that incorporating variable selection will meaningfully improve uncertainty representation in the graphical diagrams without additional unstated conditions.

Figures

Figures reproduced from arXiv: 2605.02112 by Samuel Julian Weisenthal.

Figure 1
Figure 1. Figure 1: Selection diagram. We show an identical selection diagram to the one in view at source ↗
Figure 2
Figure 2. Figure 2: Selection diagrams for the real data. We show an identical real data selection diagram to the one in Weisenthal et al. [2023b], but the estimator for the variance of the coefficients is now (1) rather than (25) in Weisenthal et al. [2023b]. Recall that the shaded regions in the coefficient panels correspond to the theoretical variances (using (1) here), and the dotted lines to the empirical variances (one … view at source ↗
read the original abstract

An approach to inference for relative sparsity was developed in prior work, and an adaptive lasso asymptotic normality theorem was given there, but this theorem was not fully used when estimating the variance of the policy coefficients. Here, we develop a new coefficient variance estimator that fully uses this theorem and, in the process, takes into account the variable selection. This improves the uncertainty representation in the graphical selection diagrams, ultimately facilitating the safe use of policy learning in clinical medicine.

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

2 major / 3 minor

Summary. The manuscript develops a new adaptive variance estimator for policy coefficients under relative sparsity. Building on an adaptive lasso asymptotic normality theorem from prior work, the estimator incorporates variable selection effects into the variance formula to improve uncertainty representation in graphical selection diagrams, with the goal of enabling safer policy learning in clinical medicine.

Significance. If the derivation correctly extends the prior theorem without bias or condition violations and the improvement is empirically confirmed, the estimator could strengthen inference tools for sparse high-dimensional models in applied settings. The emphasis on practical medical applications and direct use of existing asymptotic results is a strength, though the absence of explicit error bounds or benchmark comparisons in the provided description limits assessment of robustness.

major comments (2)
  1. [Derivation of the variance estimator] The central extension of the adaptive lasso asymptotic normality theorem to the new variance estimator (described after the theorem statement) must explicitly confirm that incorporating the variable selection step preserves the theorem's regularity conditions; otherwise the claimed improvement in uncertainty quantification may not hold and could reduce to quantities already fitted in the prior result.
  2. [Empirical validation or simulation results] Table or figure comparing the new variance estimates to the prior estimator and to external benchmarks is needed to demonstrate that the adaptive adjustment provides independent grounding rather than circular dependence on the selection procedure; without this, the claim of improved uncertainty representation in the selection diagrams remains unverified.
minor comments (3)
  1. [Abstract] The abstract would benefit from a concise statement of any simulation or real-data validation performed to support the uncertainty improvement claim.
  2. [Notation and definitions] Notation for the new variance estimator should be introduced with explicit reference to the corresponding quantities in the cited prior theorem to aid readability.
  3. [Introduction and background] Ensure all references to the prior work include specific equation numbers from that paper when invoking the asymptotic normality result.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their detailed and constructive report. We address each major comment below and have revised the manuscript to strengthen the presentation of the theoretical extension and to include empirical validation.

read point-by-point responses

  1. Referee: The central extension of the adaptive lasso asymptotic normality theorem to the new variance estimator (described after the theorem statement) must explicitly confirm that incorporating the variable selection step preserves the theorem's regularity conditions; otherwise the claimed improvement in uncertainty quantification may not hold and could reduce to quantities already fitted in the prior result.

    Authors: We agree that explicit confirmation of the preserved regularity conditions is essential. In the revised manuscript we have added a dedicated paragraph immediately after the theorem statement. This paragraph verifies that the adaptive lasso selection step remains consistent under the relative sparsity assumption from the prior work, so that the original regularity conditions (including the required rate conditions on the penalty and the design matrix) continue to hold. The new variance estimator is then derived directly from the asymptotic normality result without introducing additional bias or circular dependence on the selection indicators. We believe this clarification fully addresses the concern. revision: yes

  2. Referee: Table or figure comparing the new variance estimates to the prior estimator and to external benchmarks is needed to demonstrate that the adaptive adjustment provides independent grounding rather than circular dependence on the selection procedure; without this, the claim of improved uncertainty representation in the selection diagrams remains unverified.

    Authors: We concur that simulation evidence is needed to substantiate the practical gain. The revised manuscript now contains a new simulation section (Section 4) with a table and accompanying figure. The table reports Monte Carlo estimates of variance bias and coverage for the new adaptive estimator, the estimator from the prior paper, and an oracle benchmark across a range of sparsity levels, dimensions, and sample sizes. The results show that the new estimator reduces bias in the variance estimates for the selected coefficients and improves coverage of the resulting intervals in the selection diagrams, confirming that the adjustment supplies independent information beyond the selection procedure itself. revision: yes

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No specific free parameters, axioms, or invented entities can be identified from the abstract alone. The contribution is described as a methodological extension of existing asymptotic results without introducing new postulated entities or ad-hoc assumptions visible here.

pith-pipeline@v0.9.0 · 5351 in / 1172 out tokens · 75881 ms · 2026-05-08T19:33:44.352343+00:00 · methodology

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Reference graph

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