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arxiv: 2606.11548 · v1 · pith:I6JWRZO4new · submitted 2026-06-10 · 📊 stat.ME

Estimating the local false discovery rate under an unknown symmetric null

Daniel Xiang , William Fithian , Nikolaos Ignatiadis , Jake A. Soloff , Asaf Weinstein This is my paper

Pith reviewed 2026-06-27 09:08 UTC · model grok-4.3

classification 📊 stat.ME
keywords local false discovery ratesymmetric nulltwo-groups modelmultiple hypothesis testingdensity ratio estimationlogistic regressionspline basisknockoff filter
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The pith

Estimating the surrogate density ratio f(-w)/f(w) yields asymptotic local false discovery rate control under a symmetric null.

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

This paper develops methods to estimate the local false discovery rate when the null distribution is symmetric about zero but otherwise unknown. In the two-groups model, the authors target the ratio of the marginal density at negative and positive points as a surrogate for the lfdr. They show that any consistent estimator of this ratio, when used to threshold tests at a nominal level, achieves asymptotic control of the lfdr for the rejected hypotheses. A logistic regression approach using natural cubic splines is proposed as a practical estimator. This approach is motivated by modern multiple testing methods like knockoffs that produce symmetric null statistics.

Core claim

In the stripped-down two-groups model with symmetric null, the local false discovery rate can be estimated by targeting the surrogate f(-w)/f(w) for w > 0, and any consistent estimator of this surrogate yields asymptotic lfdr control for the procedure that thresholds at the nominal level.

What carries the argument

The surrogate density ratio f(-w)/f(w) for positive w, which serves as a proxy for the lfdr and is estimated using logistic regression with natural cubic spline basis functions.

If this is right

  • Thresholding the estimated surrogate at level alpha produces a procedure with asymptotic lfdr control at alpha.
  • The control property requires no knowledge of the null density beyond its symmetry about zero.
  • The logistic regression estimator with splines provides one route to the needed consistency.
  • The framework applies directly to variable selection problems that transform scores into statistics with symmetric nulls.
  • It extends earlier lfdr analysis under known null to the unknown-null case while preserving the control guarantee.

Where Pith is reading between the lines

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

  • The same surrogate idea could be adapted to other multiple-testing procedures that enforce null symmetry.
  • Pairing the estimator with knockoff-based selection might tighten local error rates in high-dimensional regression.
  • Empirical checks on data sets where symmetry is engineered but the null shape is unknown would test finite-sample behavior.
  • Relaxing independence assumptions while retaining symmetry might be a natural next direction.

Load-bearing premise

The null distribution of the test statistics is symmetric about zero.

What would settle it

Simulations in which symmetry holds and a provably consistent estimator of the surrogate is applied, yet the empirical local false discovery proportion among rejections fails to converge to the nominal threshold as sample size increases.

Figures

Figures reproduced from arXiv: 2606.11548 by Asaf Weinstein, Daniel Xiang, Jake A. Soloff, Nikolaos Ignatiadis, William Fithian.

Figure 1
Figure 1. Figure 1: HIV drug discovery example. The left panel shows the Knockoffs rejection set targeting 20% [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Left: Illustration of the histogram-based estimates on HIV data. The estimate of clar( [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The slopes of secant and tangent lines correspond to the right-hand sides of ( [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Illustration of two decompositions of the marginal density [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Comparison of clar estimates for several methods in the Gaussian two-groups model with location [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: On the left, a histogram of Lasso-sign-max (LSM) importance statistics is shown, obtained from [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Illustration of the cubic spline logistic method for clar estimation in a knockoffs setting. To [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Standard errors for the independent two-groups example. [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Standard errors for the knockoffs example. Left panel shows parametric bootstrap estimates of [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The left panel shows a histogram of log2 competition ratios from the protein analysis in Byun et al. (2023), where the histogram to the left of zero is reflected to the positive real line. In the right panel, we show two estimates of clar in the protein example: one using a rough histogram estimator (blue), and the other a natural cubic spline logistic regression estimator (black). assume (i) that log2 CR… view at source ↗
Figure 11
Figure 11. Figure 11: Left: histogram of pooled W-statistics after dividing by maxj |Wj | for each protease inhibitor drug (APV, NFV, ATV, LPV, IDV, SQV, RTV), with the negative tail mirrored onto the positive axis (blue curve). Right: pooled clar estimate across drugs after rescaling by the maximum absolute value |W| within each drug class. Orange triangles show the empirical FDP within bins of roughly 0.025 units on the norm… view at source ↗
Figure 12
Figure 12. Figure 12: The above plots assess convergence of the empirical cdfs for three types of [PITH_FULL_IMAGE:figures/full_fig_p030_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: The above plots assess convergence of the empirical cdfs restricted to nulls for three types of [PITH_FULL_IMAGE:figures/full_fig_p031_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Simulation with delta method CIs. Left: empirical coverage rate over 1000 resamplings of the [PITH_FULL_IMAGE:figures/full_fig_p032_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Histograms of the W-statistics used to esetimate the clar. For the independent case, the entries of X are i.i.d. N(0, 1) random variables. For the correlated case, each row of X is generated from a multivariate normal distribution with toeplitz covariance matrix (1 along the diagonal, 0.1 adjacent to the diagonal, 0.1 2 two away from the diagonal, etc.). The bottom row shows the W-statistic distribution u… view at source ↗
Figure 16
Figure 16. Figure 16: Repeat the experiment in the middle panel of [PITH_FULL_IMAGE:figures/full_fig_p034_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Plotted above are the clar estimates with ground truth labels (obtained from the TSM list) along [PITH_FULL_IMAGE:figures/full_fig_p035_17.png] view at source ↗
read the original abstract

This paper is concerned with estimating the local false discovery rate (lfdr) in a two-groups model where the only assumption regarding the null distribution is symmetry about zero. Our motivation comes from the contemporary framework for multiple hypothesis testing, particularly relevant in variable selection problems, which transforms any user-specified scores into statistics whose null distributions are symmetric about zero, whereas enrichment to the right of zero is generally expected for the non-nulls. While modern methods such as the knockoff filter (Barber and Candes; 2015) are able to exploit the null property for controlling the false discovery rate (FDR), an arguably more appropriate goal is to target control of the local false discovery rate for the rejected hypotheses, as proposed in Soloff et al. (2024) where the standard two-groups model (known $f_0$ and independence) is analyzed. Here, we take a step in this direction and propose to estimate the lfdr by targeting the surrogate density ratio $f(-w)/f(w)$, for $w>0$, where $f$ is the marginal density in the aforementioned ``stripped-down'' two-groups model. We study several estimators and propose a logistic regression based method with natural cubic spline basis. We also show that any consistent estimator of this surrogate yields asymptotic lfdr control of the multiple testing procedure that thresholds the estimate at the nominal level.

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 paper proposes estimating the local false discovery rate (lfdr) in a two-groups model under the sole assumption that the null distribution is symmetric about zero. It targets the surrogate ratio r(w) = f(-w)/f(w) for w > 0 (where f is the marginal density) rather than the lfdr directly, studies several estimators, and introduces a logistic regression estimator using natural cubic splines. The central claim is that any consistent estimator of this surrogate yields asymptotic lfdr control for the procedure that rejects when the estimated ratio falls below the nominal level.

Significance. If the asymptotic control result can be established rigorously, the contribution would be significant for multiple testing in settings such as knockoff-based variable selection, where only symmetry of the null is available. The surrogate approach avoids needing a fully specified null density, and the spline-based logistic estimator provides a practical, implementable method. The general consistency claim, if properly qualified, would broaden applicability beyond fully parametric nulls.

major comments (1)
  1. [Abstract] Abstract (central claim on asymptotic lfdr control): The assertion that 'any consistent estimator' of the surrogate r(w) = f(-w)/f(w) yields asymptotic lfdr control for the data-dependent thresholding procedure is load-bearing but appears incomplete. Pointwise consistency in probability at fixed w does not automatically guarantee that the proportion of rejected hypotheses with true lfdr exceeding the nominal level vanishes asymptotically, because the rejection set is random and depends on the estimator near the threshold. A precise statement of the required convergence mode (e.g., uniform consistency on a neighborhood of the cutoff or convergence in a norm controlling symmetric differences of level sets) together with the corresponding proof would be needed to substantiate the result.
minor comments (2)
  1. [Abstract] The abstract states that 'several estimators' are studied before proposing the logistic spline method, but provides no enumeration or comparison; adding a brief list or reference to the relevant section would improve clarity.
  2. The connection between the proposed surrogate and the lfdr control result in Soloff et al. (2024) is referenced but not expanded; a short paragraph contrasting the known-null case with the symmetry-only case would help readers follow the extension.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful and constructive review. The comment on the asymptotic lfdr control claim is well-taken and highlights an important technical point. We address it directly below and will revise the manuscript to strengthen the statement and supporting arguments.

read point-by-point responses

  1. Referee: The assertion that 'any consistent estimator' of the surrogate r(w) = f(-w)/f(w) yields asymptotic lfdr control for the data-dependent thresholding procedure is load-bearing but appears incomplete. Pointwise consistency in probability at fixed w does not automatically guarantee that the proportion of rejected hypotheses with true lfdr exceeding the nominal level vanishes asymptotically, because the rejection set is random and depends on the estimator near the threshold. A precise statement of the required convergence mode (e.g., uniform consistency on a neighborhood of the cutoff or convergence in a norm controlling symmetric differences of level sets) together with the corresponding proof would be needed to substantiate the result.

    Authors: We agree that the abstract statement is imprecise and that pointwise consistency alone is insufficient to control the random rejection set. The manuscript's theorem assumes consistency of the estimator in a mode that ensures the measure of the symmetric difference between the estimated and oracle level sets vanishes in probability (implicit in the proof via the continuous mapping theorem applied to the thresholded indicator). However, this mode is not stated explicitly in the abstract or highlighted in the main text. In the revision we will (i) replace the abstract claim with a precise statement requiring uniform consistency on compact neighborhoods of the threshold or L1 convergence of the level-set indicators, (ii) add a short paragraph in Section 3.3 spelling out the required convergence mode, and (iii) expand the proof sketch in the supplement to verify that the proportion of false rejections vanishes. These changes will be made without altering the core result. revision: yes

Circularity Check

0 steps flagged

Minor self-citation to prior work; central claim on surrogate consistency is independent

full rationale

The surrogate r(w) = f(-w)/f(w) is defined directly from the marginal density in the stripped-down two-groups model and is independent of the lfdr quantity. The paper states that it shows any consistent estimator of this surrogate yields asymptotic lfdr control, presenting this as a derivation within the current manuscript rather than reducing it to a prior result by construction. The citation to Soloff et al. (2024) supports only the motivation and the known-f0 case, not the load-bearing step for the unknown-null extension here. No self-definitional, fitted-input, or ansatz-smuggling patterns appear in the provided claims or equations.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the two-groups model with symmetry of the null as the sole modeling assumption; no free parameters, invented entities, or additional axioms are mentioned in the abstract.

axioms (1)
  • domain assumption The null distribution is symmetric about zero in the two-groups model.
    Explicitly stated as the only assumption regarding the null distribution.

pith-pipeline@v0.9.1-grok · 5788 in / 1047 out tokens · 23271 ms · 2026-06-27T09:08:12.333093+00:00 · methodology

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Reference graph

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