Pith. sign in

REVIEW 3 major objections 5 minor 6 references

Asymptotic Uniform False Discovery Rate Control for Inference of Time-varying Correlations

T0 review · 3 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper proves that false discovery rate control can be enforced uniformly over the entire time range when constructing time-varying correlation networks from high-dimensional, locally stationary, non-Gaussian time series with jumps in t

desk verdict Genuine extension of the FWER work to uniform FDR control, but the load-bearing Gaussian approximation lemma is not actually proved here. read the letter →

arxiv 2512.10467 v3 pith:7O7Y43RY submitted 2025-12-11 stat.ME econ.EMmath.STstat.TH

Bufan Li , Lujia Bai , Weichi Wu This is my paper · ORCID
classification stat.MEecon.EMmath.STstat.TH MSC 62H1562M1062F40
keywords time-varyingcorrelationnetworkfalsediscoveryratelocallystationarytimeseriesmultiplierbootstrapGaussianapproximationnon-stationarychangepointshigh-dimensionalinference
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper aims to show that false discovery rate (FDR) control can be achieved simultaneously across all time points when learning time-varying correlation networks from high-dimensional, locally stationary, possibly non-Gaussian time series with jumps in their means. It constructs time-varying P-values for every pair of series at every time point using a multiplier bootstrap applied to difference-based estimates of the time-varying correlation. It then proves that applying the Benjamini–Yekutieli procedure (or the Benjamini–Hochberg procedure under a positive-dependence condition) keeps the FDR below the target level uniformly over time. If correct, this gives practitioners a principled way to threshold correlation networks at arbitrary time points without losing error control.

What carries the argument

The load-bearing object is the difference-based local-linear estimator of the time-varying covariance, reduced to a weighted moving average of an innovation process Ξ_{j,i,l}. A new hyperrectangle Gaussian approximation shows that the joint distribution of the standardized deviations can be replaced by a Gaussian vector with the same autocovariance structure; the multiplier bootstrap replicates that Gaussian vector. The resulting time-varying P-values feed directly into the B-H and B-Y thresholds.

What would settle it

Simulate a locally stationary Gaussian process with zero true correlations at all times and known jump locations, run Algorithm 1 with the B-Y procedure at level α over a dense grid of t, and check whether the empirical sup_t FDP exceeds α (or the paper's bound) as n grows. A more direct check is to compare the empirical distribution of the bootstrap maximum deviation to the Gaussian approximation in the proof of Lemma 3 for a simple AR(1) model; if the Kolmogorov distance decays slower than (nb)^{-1/8}(log n)^4, the proof fails.

Watch

Extended reading notes

Core claim

The central claim is that the vector of bootstrap statistics across all coordinate pairs and time points is asymptotically distributed like a high-dimensional Gaussian process whose hyperrectangle probabilities match those of the standardized correlation estimates, making the time-varying P-values simultaneously valid. Consequently, Theorem 4 states that for the Benjamini–Yekutieli procedure, lim_{n→∞} sup_{t∈[b,1−b]} [FDR(t) − 2α|H0(t)|/(p(p−1))] ≤ 0, and Theorem 5 gives the same for the Benjamini–Hochberg procedure under a PRDS assumption. The guarantees hold with the dimension p growing polynomially in the sample size n.

Load-bearing premise

The whole argument leans on an unproven hyperrectangular Gaussian approximation rate (Lemma 3, adapted from a prior work with 'minor modifications', and Lemma 1 stated without proof); if that rate fails, the simultaneous validity of the P-values—and with it both FDR theorems—collapses.

Editorial extensions

If this is right

  • At any time t, the network edge set can be read off the B-Y threshold with a uniform guarantee that the expected proportion of false edges is at most α times the null proportion, up to asymptotically negligible terms.
  • Because the guarantee is uniform in t, an analyst can scan the whole time interval for edges without a multiple-comparison penalty that grows with the number of time points examined.
  • The B-H variant, valid under positive dependence, gives FDP close to the target with lower false negative proportion; B-Y is more conservative but safe under arbitrary dependence.
  • True edges are detected with probability tending to 1: P-values for alternatives tend to 0.
  • The framework tolerates non-Gaussian, nonlinear, locally stationary errors with jumps in the mean, so it applies to EEG and financial returns.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: the uniform-in-time guarantee means the network can be treated as a continuous object; one could in principle threshold the estimated correlation surface at every time and report the resulting dynamic graph without re-adjusting α.
  • Editorial inference: the bootstrap P-value construction is modular; the same hyperrectangle Gaussian approximation should extend to lagged (directed) correlation networks, which the paper only sketches, and possibly to partial correlations, a direction the authors list as future work.
  • Editorial inference: a natural stress test is to run the method on data with all null correlations and record sup_t FDP; if the B-Y bound is loose, one could improve power by replacing the log(p∗)+γ penalty with a data-dependent dependence-aware threshold.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper develops a bootstrap-based procedure for testing time-varying pairwise correlations in high-dimensional locally stationary time series with possible jumps in the means, and proves uniform-in-time asymptotic false discovery rate control for the resulting time-varying correlation networks. The main theoretical engine is a hyperrectangular Gaussian approximation for the product sequence of two locally stationary processes (Lemma 3), which is combined with a multiplier bootstrap to produce simultaneously valid time-varying P-values. Theorems 4 and 5 then apply Benjamini–Yekutieli and Benjamini–Hochberg procedures to these dependent P-values and establish sup_t FDR control. The paper also contains numerical simulations and applications to EEG and financial data.

Significance. If the main theorems are correct, the paper makes a useful contribution: it extends earlier FWER-based inference for time-varying correlation networks (Bai and Wu, 2025) to FDR control, under a broad model class that allows non-Gaussianity, non-stationarity, and abrupt mean changes. The proposed procedure is clearly described, and the simulation and real-data analyses are informative. A particular strength is that the final FDR-control argument is transparent and uses standard B-Y/B-H calculations. However, the central theoretical claim is conditional on a high-dimensional Gaussian approximation (Lemma 3) whose proof is only sketched by invoking an author-overlapping preprint, and on Lemma 1 whose proof is omitted. These are load-bearing technical premises, not peripheral details.

major comments (3)
  1. [Abstract vs. §1 and §4 (Theorems 4–5)] The arXiv abstract defines AuFDR as the expectation of the L_r-norm of the FDP process with r allowed to diverge, but this quantity does not appear anywhere in the main text. Theorems 4 and 5 instead prove sup_{t∈[b,1-b]} [FDR(t) − 2α|H0(t)|/(p(p−1))] ≤ 0. These are different criteria: control of the time-wise supremum of pointwise FDR is not the same as control of an L_r norm (even for diverging r). The authors should either define AuFDR in the body and prove a theorem for it, or remove the AuFDR claim from the abstract/title. This is a mismatch in the paper's advertised central contribution.
  2. [Appendix E.3, Lemma 3] Lemma 3 is the key Gaussian approximation used in Theorem 1, and hence in Theorems 3–5. Its proof is not self-contained: equation (28) is asserted to follow from Propositions 6–7 of Wu et al. (2024) with “minor modifications” after a “very careful investigation”, but two of the modifications are explicitly non-minor: Lemma 2(iii) gives δ_q(H_{i,l,Ξ},r)=O(χ^{r−h}) rather than O(χ^r), and the tail condition is sub-Weibull rather than exponential. These changes introduce the terms (φ^q n p^* χ^{(M−h)q})^{1/(1+q)} and exp(−tM_x^κ) in (28), but the subsequent optimization in M, M_x, M_y, φ, β that yields the claimed O((nb)^{−1/8}(log n)^4) bound is not shown. There is also no explicit accounting of how the dimension p enters the final sup over hyperrectangles. Since the FDR theorems require p^2(θ_n^{(1)} + θ_n^{(2)}) → 0, an uncounted polynomial-in-p factor in Lemma 3 would invalidate the mai
  3. [Appendix E.1, Lemma 1] Lemma 1 is used in the proof of Theorem 1 to pass from the discretized maximum to the supremum over t and to compare the original statistic with the Gaussian proxy (see equations (20)–(21) and subsequent steps). The lemma is stated with the sentence “For simplicity we omit the proof” and a citation to Lemma S1 of Dette and Wu (2021). This is not adequate for a lemma on which the main theorem depends. The authors should provide a full proof or state the exact Dette–Wu lemma with the precise adaptation and verify its hypotheses in this setting.
minor comments (5)
  1. [Throughout] There are two different titles/abstracts: the arXiv version advertises “Asymptotic Uniform False Discovery Rate Control for Inference of Time-varying Correlations” and an L_r-based AuFDR, while the main text is titled “Learning Time-Varying Correlation Networks with FDR Control via Time-varying P-values” and uses a sup-FDR criterion. Please harmonize these.
  2. [§1] Typo: “investiated” should be “investigated”; elsewhere “guaranties” should be “guarantees” and “provides theoretical guarantees for the B-H and B-Y procedure” is repeated with a grammatical error in the sentence before Theorem 4.
  3. [Appendix E.1, Lemma 1] The statement of Lemma 1 begins with “for every y∈R^{p*} and δ>0” followed by a supremum over y; the first “for every y” is redundant or misstated. Please clarify the quantifiers.
  4. [Algorithm 1, step 6] The denominator for the bootstrap statistic is √(2w⌈nb⌉), where b = max_{i,l} b_{i,l}, while the numerator is computed with a sum over 2⌈nb_{i,l}⌉−w. If the use of the global b is intentional due to the scaling c_{i,l} in (8), please state this explicitly; otherwise this appears to be a typo.
  5. [§4, Assumption 2 paragraph] The text says Assumption 2 is satisfied for sufficiently large q if ω≍n^{2/5}; the parameter ω is not defined. It is presumably the block size w. Please correct the notation.

Circularity Check

1 steps flagged · score 4.0 of 10

FDR control is not constructed from its inputs, but its load-bearing Gaussian-approximation lemma is imported from same-author prior work with unproved modifications.

  1. self citation load bearing [Appendix E.3, Lemma 3 proof; used in Theorem 1 (Section 4)]
    "With a very careful investigation of the proof of Proposition 6 and 7 of Wu et al. (2024), we have equation (236) still holds in view of results in Lemma 2 and Assumption 1 with minor modifications, which is, for some small enough but positive constant t, |E{m(ΞG,W )−m(ZG,W,(M) )}|≲(φ2+φβ)(M−1x+M−1y)+..."

    The uniform-FDR theorem chain is: Theorems 4–5 use Theorem 3 for simultaneous P-value validity; Theorem 3 uses Theorem 1; and Theorem 1's proof invokes 'Lemma 3 implies ...' at the key Gaussian-approximation step. Lemma 3 is therefore load-bearing. Its proof is not supplied in this manuscript; it is asserted to follow from Propositions 6–7 of Wu et al. (2024) after 'minor modifications' that include changing the dependence decay to O(χ^{r−h}) and sub-Weibull tails. Since Wu et al. (2024) shares an author with the present paper and the modified result is not independently established, the new FDR conclusion rests on a same-author citation rather than on a self-contained derivation of the modified approximation.

full rationale

The central FDR-control claim is not circular by construction: Theorems 4 and 5 apply the external Benjamini–Yekutieli and Benjamini–Hochberg theorems to the produced P-values, and the target 'FDR(t) ≤ 2α|H0(t)|/p(p−1)' is not used as an assumption anywhere. The bootstrap P-values are derived from a difference-based estimator and multiplier bootstrap, not fitted to the FDR target. However, the proof chain that validates those P-values reduces at its technical core to Lemma 3, a hyperrectangular Gaussian approximation whose proof is delegated to same-author prior work with unstated 'minor modifications'. Lemma 1, also needed for Theorem 1, is stated with its proof omitted ('For simplicity we omit the proof'). These are proof-completeness and self-citation concerns rather than definitional circularity; the FDR theorem still has substantial independent content via the external B-Y/B-H machinery. Score 4 reflects a load-bearing same-author citation at the base of the derived P-value validity, without claiming the whole derivation is equivalent to its inputs.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new physical or statistical entities beyond the method itself. The abstract's AuFDR criterion is introduced but never used in the body, so it is not an operational invention. The axioms are mostly standard assumptions for locally stationary processes; the main additional load is the unproved Gaussian approximation lemma borrowed from an author-overlapping preprint.

free parameters (4)
  • lag h = h = ceil(2 log n) recommended
    Theory requires h ~ log n; the paper recommends h = ceil(2 log n) based on simulations. The central estimator uses h to difference the data.
  • bandwidths b_{i,l} = selected via GCV, with b = max b_{i,l} ~ n^{-1/5} in theory
    Local linear smoothing of the differenced products requires bandwidth choices satisfying nb^4 -> inf and nb^6 -> 0. Chosen data-adaptively, not fitted to the FDR target.
  • smoothing parameters m_{i,l}, eta for variance estimator = selected via minimum-volatility method; m ~ n^{2/7}, eta ~ n^{-1/7} in theory
    Used to estimate the long-run variance Gamma^2. Tuning is data-driven and not part of the FDR claim itself.
  • block size w for bootstrap = selected via minimum-volatility method
    Multiplier bootstrap uses block sums of estimated innovations; w enters the rate theta_n and must satisfy conditions in Assumption 2.
assumptions (6)
  • domain assumption Assumption 1(i)-(iii): locally stationary error process with L_{2q} stochastic Lipschitz, sub-Weibull tails, and geometric physical dependence decay.
    These structural conditions define the allowed data-generating processes and are used throughout Theorems 1-3.
  • domain assumption Assumption 1(iv)-(v): smoothness and uniform boundedness away from zero of gamma_{i,l} and Gamma_{i,l}.
    Needed for the variance estimator to be well-behaved and for the Gaussian approximation to apply to the standardized statistic.
  • domain assumption Assumption 2: bandwidth and scaling conditions, e.g., p^2/(n^{q/2+phi} b^{q/2-1} h + ...) -> 0.
    These rates ensure the Gaussian approximation and bootstrap errors vanish; they restrict how fast p can grow with n.
  • domain assumption Assumption 3 (PRDS): the P-values satisfy positive regression dependence on a subset.
    Required for the B-H procedure (Theorem 5). The paper notes PRDS is hard to verify and relies on simulations for robustness.
  • ad hoc to paper Lemma 3: hyperrectangular Gaussian approximation for the product sequence, adapted from Propositions 6-7 of Wu et al. (2024).
    The key approximation is not fully proven in this paper; it is described as a 'minor modification' of an author-overlapping preprint. The FDR control theorems depend on it.
  • standard math Lemma 1: error-transfer lemma for sup-norm Gaussian approximation; proof omitted.
    Used to convert approximation errors in L_q norm to sup-norm CDF differences; the paper states the proof is omitted, citing Dette and Wu (2021).

how reviews work

0 comments
Cite this review

Pith. "Pith review of Asymptotic Uniform False Discovery Rate Control for Inference of Time-varying Correlations." pith.science (2026). https://pith.science/paper/7O7Y43RY

@misc{pith2026251210467,
  author       = {Pith},
  title        = {Pith review of: Asymptotic Uniform False Discovery Rate Control for Inference of Time-varying Correlations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7O7Y43RY}},
  note         = {Machine review of arXiv:2512.10467}
}
abstract

Inference for locally stationary time series is challenging because the associated hypotheses form an uncountable collection over a continuous time interval, making pointwise false discovery rate (FDR) control inadequate for simultaneous statistical guarantees. We introduce a novel asymptotically uniform false discovery rate (AuFDR), defined as the expectation of the $L_r$-norm of the false discovery proportion (FDP) process where $r$ is allowed to diverge, to quantify and control false discoveries uniformly over time. To operationalize AuFDR control, we develop an inferential framework for time-varying correlations in high-dimensional nonstationary time series that allows for non-Gaussianity, nonlinearity and possible jumps in mean functions. The proposed approach combines robust difference-based estimators with a multiplier-bootstrap procedure to construct uniformly valid time-varying $P$-values. Based on these $P$-values, we propose a time-varying Benjamini--Yekutieli procedure for controlling the AuFDR under arbitrary dependence and establish its asymptotic validity. Extensive simulations demonstrate the finite-sample performance of the proposed method in controlling the AuFDR. Applications to EEG data and financial time-series data illustrate its practical utility.

Figures

Figures reproduced from arXiv: 2512.10467 by the authors.

Figure 1
Figure 1. Top: ERP signals from three (out of p = 64) electrodes in one subject, measured at n = 256 time points. Red areas mark three selected time points. Bottom: Time￾varying correlation networks at each interval; nodes are electrodes, edges indicate significant correlations. At the third interval, Electrode 4 is disconnected due to the lack of significant correlation [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Constructing multiple lags correlation networks (for example, [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Time-varying FDP and FNP of Case 1 (averaged across 100 repetitions). [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Time-varying connection proportions (averaged across subjects in each group) [PITH_FULL_IMAGE:figures/full_fig_p016_4.png]
Figure 5
Figure 5. Figure 5: Correlation network of WRDS data from 2011 to 2018 [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

6 extracted references · 1 linked inside Pith

  1. [1]

    M., Ozdaglar, A., and Tahbaz-Salehi, A

    Acemoglu, D., Carvalho, V. M., Ozdaglar, A., and Tahbaz-Salehi, A. (2012). The network origins of aggregate fluctuations.Econometrica, 80(5):1977–2016. Bai, L. and Wu, W. (2025). Uniform variance reduced simultaneous inference of time-varying correlation networks.IEEE Transactions on Information Theory, pages 1–1. Bailey, N., Pesaran, M. H., and Smith, L....

    2012

  2. [254]

    and Cheng, G

    Zhang, X. and Cheng, G. (2018). Gaussian approximation for high dimensional vector under physical dependence.Bernoulli, 24(4A):2640 –

    2018

  3. [534]

    Barber, R. F. and Candès, E. J. (2015). Controlling the false discovery rate via knockoffs. The Annals of statistics, pages 2055–2085. Basu, S. and Subba Rao, S. (2023). Graphical models for nonstationary time series.The Annals of Statistics, 51(4):1453–1483. Benjamini, Y. and Hochberg, Y. (1995). Controlling the false discovery rate: a practical and powe...

    2015

  4. [1044]

    De Domenico, F., Livan, G., Montagna, G., and Nicrosini, O. (2023). Modeling and simulation of financial returns under non-gaussian distributions.Physica A: Statistical Mechanics and its Applications, 622:128886. Dette, H. and Wu, W. (2021). Confidence surfaces for the mean of locally stationary functional time series.arXiv preprint arXiv:2109.03641. Dieb...

    arXiv 2023

  5. [2352]

    35 Dahlhaus, R., Richter, S., and Wu, W. B. (2019). Towards a general theory for nonlinear locally stationary processes.Bernoulli, 25(2):1013 –

    2019

  6. [2675]

    L., Begleiter, H., Porjesz, B., and Litke, A

    Zhang, X. L., Begleiter, H., Porjesz, B., and Litke, A. (1997). Electrophysiological evidence of memory impairment in alcoholic patients.Biological Psychiatry, 42(12):1157–1171. Zhao, Z. (2015). Inference for local autocorrelations in locally stationary models.Journal of Business & Economic Statistics, 33(2):296–306. Zrnic, T., Ramdas, A., and Jordan, M. ...

    1997

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.