False Discovery Rate for Functional Data

Alessia Pini; Niels Lundtorp Olsen; Simone Vantini

REVIEW 3 major objections 4 minor 17 references

False Discovery Rate for Functional Data

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper proves that a continuous Benjamini-Hochberg procedure controls the functional false discovery rate when the p-value function is positively regression dependent and the finite-grid approximation is regular.

desk verdict The fBH procedure is a genuine extension of BH to functional data with a credible control proof, but the theory only applies under an infinite-dimensional PRDS assumption that the applications do not check. read the letter →

arxiv 1908.05272 v1 pith:WFPNHN7Q submitted 2019-08-13 stat.ME

Niels Lundtorp Olsen , Alessia Pini , Simone Vantini This is my paper · ORCID

classification stat.ME

keywords falsediscoveryratefunctionaldataanalysislocalinferenceBenjamini-Hochbergproceduremultipletestingpositiveregressiondependenceadjustedp-valuefunctionclimatetemperature

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 addresses local inference for functional data, where a null hypothesis is tested at every point of a continuous domain, so there are infinitely many tests. It defines the functional false discovery rate as the expected proportion of the rejected region that is falsely rejected, measured by a weight on the domain, and introduces a continuous version of the Benjamini-Hochberg procedure: reject the points whose p-value falls below an adjusted threshold built from the measure of the p-value sublevel set. The central result is that, under positive regression dependence on the true-null region and mild regularity of the p-value function, this functional BH procedure controls the functional FDR at level $\alpha\nu(U)/\nu(T)$, where $U$ is the true-null region. Because the procedure can be approximated by ordinary BH on a fine grid, it gives practitioners a practical way to correct for multiple testing over curves and surfaces. The simulations and an application to satellite temperature data illustrate the method and show that large significant regions remain after adjustment.

What carries the argument

The load-bearing object is the cumulated p-value function $a(r)=\nu(\{s:p(s)\le r\})$, which compresses infinitely many pointwise tests into one sublevel-set curve. The fBH threshold is the largest crossing of $a(r)$ with the line $r/\alpha$, and the procedure rejects every point whose p-value lies below that crossing. The convergence proof works by applying ordinary weighted BH to $p$ evaluated on a dense uniform grid $S_k$, obtaining a discrete false-discovery proportion $Q_k$ that converges almost surely to the functional $Q$, so the finite-sample PRDS bound can be passed through the limit by dominated convergence.

What would settle it

Generate functional data on a one-dimensional domain with a known true-null region $U$, using pointwise test statistics from a Gaussian process with nonnegative covariance so the PRDS assumption is satisfied, and construct the p-value function accordingly. Apply the fBH procedure at level $\alpha$ on a sequence of increasingly fine uniform grids; across many replications the empirical functional false discovery proportion should not exceed $\alpha\nu(U)/\nu(T)$ if Theorem 3.9 is correct.

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Extended reading notes

Core claim

On a domain $T$ with measure $\nu$, the paper defines $V$ as the set of points where a true null is rejected, $S$ as the set where a false null is rejected, and the functional false discovery rate as $\mathrm{fFDR}=\mathbb{E}[\nu(V)/(\nu(V\cup S))]$. The functional BH procedure rejects $H_0^t$ at all points $t$ with $p(t)\le \alpha^*$, where $\alpha^*$ is the largest $r$ such that the cumulated p-value function $a(r)=\nu(\{s:p(s)\le r\})$ satisfies $a(r)/\nu(T)\ge r/\alpha$. Theorem 3.9 states that if the p-value function is PRDS on the true-null set $U$ and the stated level-set regularity conditions hold with probability one, then $\mathrm{fFDR}\le \alpha\,\nu(U)/\nu(T)\le \alpha$. Equivalently, the adjusted p-value function $\tilde p(t)=\min_{s\ge p(t)} \nu(T)s/\nu(\{p\le s\})$ yields the same rejection region for every $\alpha$.

Load-bearing premise

The guarantee depends on the pointwise p-value function being positively regression dependent on the true-null region, so that raising a true-null p-value can never make rejection of the set easier.

Editorial extensions

If this is right

A dense-grid application of ordinary BH converges to the infinite-domain fBH threshold, so local inference on curves and surfaces can be FDR-adjusted with standard software.
The adjusted p-value function lets a researcher compute significance after correction for every $\alpha$ at once, rather than re-running the procedure threshold by threshold.
Because fBH controls FDR rather than FWER, it is more sensitive than methods such as Fmax in the simulations, at the cost of only weak family-wise control.
With a weighted measure such as $\cos(\text{latitude})$ on the sphere, the procedure adapts to non-uniform domains; in the climate application 15.0% of Earth's area remains significant at the 5% level after adjustment, compared with 32.4% before.

Reading between the lines

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

A useful next step would be a practical diagnostic for the PRDS assumption, since the paper's proof assumes it but does not check it in the simulations or the climate application.
The same sublevel-set threshold construction could define functional analogues of weighted FDR, positive FDR, or q-value curves, giving a continuous-data counterpart to standard FDR-related quantities.
The convergence result suggests a grid-selection rule: keep refining the grid until the fBH threshold and the measure of the rejection region stop changing, since the proof requires uniform approximation of p-value level sets.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Dear colleague,

You should know two things. First, this paper genuinely extends BH to an infinite-dimensional functional setting: it defines fFDR, a continuous fBH procedure, and an adjusted p-value function, and proves that the continuous procedure is the limit of finite-grid BH under dominated convergence. That is a real contribution, and the proof strategy—reducing to Benjamini-Yekutieli on grids—is sound. Second, the theory rests on an infinite-dimensional PRDS assumption that the paper never verifies in simulations or in the climate application, so the headline guarantee is conditional.

What's new: the definition of fFDR as the expected proportion of ν-measure of false discoveries, the fBH threshold via the cumulated p-value function, and the adjusted p-value function that lets you read off significance for any α. The limit argument is careful about assumptions (a1)-(a3) and grid approximation; the appendix fills in the details. I believe Theorem 3.9 is correct as stated. The simulations show the expected behavior, including FDR control under the scenarios tested.

Where it's soft: the PRDS assumption is the load-bearing condition, and it is simply asserted for the applications. The climate data uses yearly averages over 25 years per grid point with a linear model, ignoring temporal autocorrelation; if the errors are autocorrelated, pointwise p-values are not uniform under the null and the theorem does not apply. That is a real gap between theory and the showcase application. Also, equations (3.5) and (3.6) in the weighted case have a scaling typo—sums over grid points divided by #S_k do not converge to integrals of f; this needs fixing. Minor: in Lemma A.5 the text says "all hypotheses are rejected" when it means "none are rejected" (#A_k = 0), a wording slip.

Whether you believe the central result: yes, the proof holds up under its assumptions. But the paper overclaims slightly by applying the method to data where the assumptions are not checked.

Who this is for: anyone doing local inference on functional or spatial data who wants an FDR analog. It deserves a serious referee, and the authors should be asked to fix the typos, add a diagnostic or at least a discussion of PRDS in the climate example, and address the temporal dependence issue.

Referee Report

3 major / 4 minor

Summary. The paper extends the Benjamini-Hochberg (BH) false discovery rate framework to functional data with a continuum of pointwise hypotheses. It defines a functional FDR via a measure-weighted proportion of false rejections, proposes a functional BH (fBH) procedure and an adjusted p-value function, and proves that, under positive regression dependence (PRDS) of the pointwise p-value process on the true-null set plus regularity conditions, fBH controls the functional FDR at level αν(U)/ν(T). The proof approximates the continuum by nested finite grids, applies the classical Benjamini-Yekutieli bound on each grid, and passes to the limit by dominated convergence. Two simulation studies (one 1D, one 2D) and an application to satellite temperature data are presented.

Significance. If the main theorem is correct, this is a useful and elegant contribution: it gives a principled, practically implementable way to control FDR for local inference on functional data, with the finite-grid approximation reducing to ordinary BH. The proof is self-contained and rests on the classical BY result rather than new model-specific machinery. The explicit assumptions are a strength, and the paper is careful to state what is needed for the guarantee. The main limitations are that the central definition is formally misstated, the load-bearing PRDS assumption is not checked in the simulations or the application, and the climate application may violate the uniformity-of-p-values assumption. These are fixable in revision and do not undermine the theoretical core.

major comments (3)

[Section 3.2, Definition 3.3 (and Eqs. (3.4), (3.7))] The definition of α* as 'arg max_r [ν({s : p(s) ≤ r})/ν(T) ≥ α^{-1} r]' is not a well-posed maximization, because the bracketed expression is a logical condition rather than a numerical objective function. The intended definition appears to be α* = sup{r ∈ [0,1] : a(r) ≥ r/α}, with the supremum attained under assumption (a2). The same correction is needed for b_k in Propositions 3.5 and 3.7. As written, the central object of the procedure is formally undefined, and Theorem 3.9 cannot be evaluated as stated.
[Sections 5.1, 5.2, and 6] The assumptions of Theorem 3.9 are not verified for the climate application. The pointwise p-values are obtained from one-sided t-tests in a linear model with 25 yearly observations per grid cell; if the errors are temporally autocorrelated, the p-values are not U(0,1) under H0, so the FDR bound does not apply. The paper also provides no diagnostic for the infinite-dimensional PRDS condition over the 64,800 grid points; Section 6 only asserts that 'a certain degree of positive association' is required. Consequently, the adjusted significance areas in Table 2 are not covered by the theorem. Please either re-frame the application as illustrative and outside the theorem's assumptions, or add checks (e.g., block/bootstrap or simulations with realistic dependence) that support the assumptions.
[Section 3.5, Theorem 3.9] The assumption that 'p(t) is PRDS wrt the set of true null hypotheses with probability one' is stated as if PRDS were a sample-path property. PRDS is a property of the joint distribution of the p-value process (Definition 2.5), so 'with probability one' is inappropriate. What the proof needs is that, for each finite grid S_k, the joint law of {p(t) : t ∈ S_k} is PRDS on S_k ∩ U. This should be restated, because the finite-grid step applies Theorem 2.6 to those finite-dimensional laws.

minor comments (4)

[Throughout] The acronym is inconsistently written as 'PDRS' in several places (e.g., Definition 2.5, Theorem 3.9, and Appendix A.1); it should be 'PRDS' uniformly.
[Appendix, Lemma A.5] The text says that 'all hypotheses are rejected eventually' when b_k = 0 and min p(t) > 0, but the displayed conclusion #A_k = 0 means no hypotheses are rejected; the correct wording is 'all hypotheses are accepted eventually'.
[Section 3.4.1] The phrase 'Qk behaves asymptotically as the functional false discovery proportion' is imprecise; it would be clearer to write 'Q_k → Q almost surely'.
[Figure 4] The dotted lines with slopes 1 and 0.717 are mentioned in the caption but not labeled in the figure; adding a legend or explicit annotation would make the comparison with the nominal bounds much easier to read.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: fFDR control is derived from the external Benjamini-Yekutieli PRDS theorem plus a limit argument.

full rationale

The central claim is self-contained. Theorem 3.9 obtains functional FDR control by defining finite-grid approximations Q_k through the ordinary BH procedure (Eq. 3.7), applying the external Benjamini-Yekutieli PRDS theorem (Theorem 2.6, [4]) to each finite grid, and passing to the limit via dominated convergence under regularity conditions (a1)-(a3) and grid approximation assumptions (3.5)-(3.6). The infinite-dimensional PRDS assumption is a direct extension of an independently established condition, not an input that already contains the target bound. No fitted parameter is later renamed as a prediction: the simulations use known generative models to evaluate performance, and the climate application computes pointwise p-values and applies the fBH adjustment without estimating the FDR bound from the data. The self-citations ([1], [12]) concern prior FWER methods used for comparison or motivation and are not load-bearing for the FDR control claim. The unverified PRDS condition for the climate data is a legitimate correctness/validity concern, but it is not circular reasoning.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central result is a theorem conditional on the PRDS and regularity assumptions. There are no fitted parameters. The main assumptions are clearly stated, but the PRDS condition is strong and unverified in the simulations and application.

assumptions (5)

standard math Benjamini and Yekutieli (2001) PRDS theorem: BH controls FDR at m0/m alpha for finite-dimensional PRDS p-values.
The proof of Theorem 3.9 applies Theorem 2.6 to each finite grid.
domain assumption The pointwise p-value process p(t) is PRDS on U, the set of true null hypotheses, with probability one.
This is the central dependency assumption (Theorem 3.9); it extends Definition 2.5 to the infinite-dimensional setting and is not verified in the simulations or application.
domain assumption The grid sequence {S_k} uniformly approximates all level sets of p and p|U with probability one (Eqs. 3.5 and 3.6).
Needed for Q_k to converge to Q; plausible for continuous p on uniform grids but not proven for the 2D simulation setup.
domain assumption The p-value function satisfies (a1) level sets have zero measure, (a2) non-tangential crossing at alpha*, and (a3) min p(t)>0 if alpha*=0.
Technical regularity needed for convergence of thresholds; reasonable for continuous p but not checked in the simulations.
domain assumption The measure nu is absolutely continuous with respect to Lebesgue with bounded strictly positive density f.
Used in the weighted version and in the spherical application with f proportional to cos(latitude).

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Cite this review

Pith. "Pith review of False Discovery Rate for Functional Data." pith.science (2026). https://pith.science/paper/WFPNHN7Q

@misc{pith2026190805272,
  author       = {Pith},
  title        = {Pith review of: False Discovery Rate for Functional Data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WFPNHN7Q}},
  note         = {Machine review of arXiv:1908.05272}
}

read the original abstract

Since Benjamini and Hochberg introduced false discovery rate (FDR) in their seminal paper, this has become a very popular approach to the multiple comparisons problem. An increasingly popular topic within functional data analysis is local inference, i.e., the continuous statistical testing of a null hypothesis along the domain. The principal issue in this topic is the infinite amount of tested hypotheses, which can be seen as an extreme case of the multiple comparisons problem. In this paper we define and discuss the notion of false discovery rate in a very general functional data setting. Moreover, a continuous version of the Benjamini-Hochberg procedure is introduced along with a definition of adjusted p-value function. Some general conditions are stated, under which the functional Benjamini-Hochberg procedure provides control of the functional FDR. Two different simulation studies are presented; the first study has a one-dimensional domain and a comparison with another state of the art method, and the second study has a planar two-dimensional domain. Finally, the proposed method is applied to satellite measurements of Earth temperature. In detail, we aim at identifying the regions of the planet where temperature has significantly increased in the last decades. After adjustment, large areas are still significant.

Figures

Figures reproduced from arXiv: 1908.05272 by the authors.

**Figure 2.** Simulation study: Familywise error rate, false discovery rate, false positive rate and [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗

**Figure 3.** Two simulated fields with increasing signal strength [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Average sensitivity, false positive rate (FPR) and FDR for different levels of [PITH_FULL_IMAGE:figures/full_fig_p017_4.png]

**Figure 5.** Figure 5: Above: Average yearly temperature change in centigrades, 1983-2007. Middle: Pointwise [PITH_FULL_IMAGE:figures/full_fig_p019_5.png]

**Figure 6.** Figure 6: Upper plot: Unadjusted p-values. Lower plot: FDR-adjusted p-values. Dashed lines [PITH_FULL_IMAGE:figures/full_fig_p020_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

, H¨ager, C

Abramowicz, K. , H¨ager, C. K. , Pini, A. , Schelin, L. , Sj¨ostedt de Luna, S. , and Vantini, S. (2018). Nonparametric inference for functional-on-scalar linear models applied to knee kinematic hop data after injury of the anterior cruciate ligament. Scandinavian Journal of Statistics

work page 2018

[2] [2]

and Hochberg, Y

Benjamini, Y. and Hochberg, Y. (1995). Controlling the false discovery rate: a practical and powerful approach to multiple testing. Journal of the royal statistical society. Series B (Methodological), 289–300

work page 1995

[3] [3]

and Hochberg, Y

Benjamini, Y. and Hochberg, Y. (1997). Multiple hypotheses testing with weights. Scan- dinavian Journal of Statistics 24, 3, 407–418

work page 1997

[4] [4]

and Yekutieli, D

Benjamini, Y. and Yekutieli, D. (2001). The control of the false discovery rate in multiple testing under dependency. Annals of statistics , 1165–1188

work page 2001

[5] [5]

and Thomas-Agnan, C.(2011)

Berlinet, A. and Thomas-Agnan, C.(2011). Reproducing kernel Hilbert spaces in probability and statistics. Springer Science & Business Media

work page 2011

[6] [6]

, Schwartzman, A., and others

Cheng, D. , Schwartzman, A., and others. (2017). Multiple testing of local maxima for detection of peaks in random ﬁelds. The Annals of Statistics 45, 2, 529–556

work page 2017

[7] [7]

D., and Tusher, V

Efron, B., Tibshirani, R., Storey, J. D., and Tusher, V. (2001). Empirical bayes analysis of a microarray experiment. Journal of the American statistical association 96, 456, 1151–1160. 21

work page 2001

[8] [8]

and Lane, D

Freedman, D. and Lane, D. (1983). A nonstochastic interpretation of reported signiﬁcance levels. J. Bus. Econ. Stat. 1, 4, 292–298

work page 1983