Gaussian and bootstrap approximations for functional principal component regression

Hyemin Yeon

REVIEW 4 major objections 5 minor 3 references

Gaussian and bootstrap approximations for functional principal component regression

T0 review · 4 major / 5 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read A suitable operator scaling of the FPCR estimator yields Gaussian and bootstrap distributional approximations for the slope function.

desk verdict Potentially important result—operator-scaled FPCR does have Gaussian and bootstrap limits—but the proof as written has a genuine basis-rotation gap in Proposition B.1 and leans on deferred lemmas, so it needs serious verification before the claim is believed. read the letter →

arxiv 2603.12518 v2 pith:3AO2UGQQ submitted 2026-03-12 math.ST stat.MEstat.TH

Hyemin Yeon This is my paper · ORCID

classification math.STstat.MEstat.TH MSC 62R1062G0962G2062G10

keywords functionalprincipalcomponentregressionscalar-on-functioncentrallimittheorembootstrapconsistencyWassersteindistanceoperatorscalinghypothesistestingforslopefunctiondataanalysis

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

This paper tries to overturn a long-standing obstacle in scalar-on-function regression: the slope estimator built from functional principal components had been shown to have no non-degenerate limit under any scalar rescaling, which blocked direct inference about the slope function. The author's claim is that rescaling by the sample covariance square root together with a J^{-1/2} factor restores valid Gaussian and residual-bootstrap approximations in the Wasserstein distance, with the number of components J growing with the sample size. If true, the classical FPCR estimator becomes usable for hypothesis tests on the full slope function, including tests based on the L2 norm and on the supremum norm. The conditions used are stated in the paper as (C1)-(C4), (E), (B_v) with v>2, and (R_w) with w>6. The load-bearing part of the proof is a bias bound whose perturbation lemmas are partly deferred to a companion manuscript and partly omitted, so the claim stands on those bounds.

What carries the argument

The mechanism is the operator scaling GammaHat_J^{1/2}, the square root of the sample covariance operator restricted to the first J eigencomponents (the pseudo-square-root of the covariance), together with the deterministic factor J^{-1/2}. In the proof the statistic is split into a variance term sqrt(n/J) GammaHat_J^{-1/2} U_n and a bias term B_n; the bias is decomposed into four pieces (D.1)-(D.4) and controlled by operator-perturbation lemmas (D.2-D.3) built from resolvent and contour-integral representations of the covariance operators. The square-root operator normalizes the empirical eigenvectors so that the variance term stabilizes to a Gaussian over the first J components, while the

What would settle it

Fix eigenvalues satisfying (C2)-(C4) with slow eigengap decay, for instance gamma_j approximately j^{-a} with a near the threshold, simulate n and J, and compare the empirical distribution of T_J with its residual-bootstrap counterpart; if the conditional Wasserstein distance fails to go to zero as n grows, the bias decomposition or the omitted Lemma D.3 bound is wrong. Alternatively, check numerically the operator bound ||GammaHat_J^{1/2} - Gamma_J^{1/2}|| against the claimed n^{-1/2} J^{3/2} (log J)^{1/2} rate used in the bias argument.

Watch

Extended reading notes

Core claim

The central claim is Theorem 2.1: under Conditions (C1)-(C4), (E), (B_v) for v>2, and (R_w) for w>6, both W_X(T_J, sigma J^{-1/2} G_J) and W_X(T_J, T*_J) converge to zero in probability, where T_J = sqrt(n/J) GammaHat_J^{1/2}(betaHat_J - beta), G_J is a centered Gaussian element with covariance the rank-J projection, and T*_J is the residual-bootstrap analogue. In words, after operator scaling the FPCR estimator has a Gaussian limit, and the bootstrap distribution is consistent for the sampling distribution. This overturns the earlier negative result that no scalar rescaling of betaHat_J can work. Theorem 3.1 then shows that, under beta = 0, bootstrap quantiles of either the L2 norm or the s

Load-bearing premise

The proof of the bias bound relies on operator perturbation lemmas whose detailed derivations are not fully present in this paper, with one bound quoted from a companion manuscript and another proof omitted, so the Gaussian and bootstrap conclusions stand only if those bounds hold under Conditions (C1)-(C4).

Editorial extensions

If this is right

Under the paper's conditions, the FPCR slope estimator has a non-degenerate Gaussian limit after operator scaling, so direct asymptotic inference on the full slope function becomes possible.
The residual bootstrap is consistent under the same assumptions used for the Gaussian approximation, and it does not require a preliminary Gaussian limit to work.
Because the Wasserstein approximation transfers through Lipschitz maps, bootstrap inference is valid for any Lipschitz functional of the scaled estimator, including the L2 and supremum-norm test statistics.
For the L2-norm statistic, the result reproduces and extends earlier quadratic-form bootstrap tests under weaker assumptions; for the supremum-norm statistic, it supplies a new testing procedure.
In the paper's simulations, the bootstrap tests keep empirical size near 0.05 while showing higher power than the benchmark methods at small signal strengths.

Reading between the lines

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

One direct extension would be to confidence regions or simultaneous bands for beta by inverting the bootstrap distribution of the operator-scaled deviations, rather than only testing beta = 0.
The same operator-scaling idea could plausibly resolve the degenerate-limit problem in related settings, such as function-on-function regression, generalized functional linear models, or multivariate and sparse functional regressors, since the degeneracy has the same source.
A cautious reader should verify the deferred perturbation bounds (Lemma D.2's quoted step and Lemma D.3's omitted proof) before relying on the theorem in applications; checking them for slowly decaying eigenvalues would settle the residual risk.

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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,

This is a paper you should put on your radar. The claimed result is genuinely new: using operator scaling rather than scalar scaling, the FPCR slope estimator is asserted to have Gaussian and bootstrap approximations in Wasserstein distance, which would overturn the long-standing no-CLT message of Cardot et al. (2007). The paper also deserves credit for separating the bootstrap consistency from the Gaussian approximation—the bootstrap half does not depend on the Gaussian half—and for setting up a clean variance/bias decomposition. The application to L2 and sup-norm tests is a natural and useful payoff.

Now the soft spots. The proof is not complete as printed. Lemma D.2 relies on an unpublished/forthcoming result by the same author (Yeon 2026+) to drop an absolute summability condition, and Lemma D.3's proof is omitted entirely. Those are addressable, but they are load-bearing: the bias term in Proposition B.4 is controlled by exactly those lemmas.

There is also a genuine gap that the stress-test note identified, and I think it lands. In Proposition B.1, the proof proves a J-dimensional multivariate CLT for the coefficients in the sample eigenbasis {φhat_j}, and then asserts W_X(√(n/J) Γhat^{-1/2} U_n, σ J^{-1/2} G_J) = J^{-1/2} W_{R^J}(√n \bar Z_J, σ Z). That equality is not valid: the target G_J lives in the population eigenbasis span{φ_j}, while the argument is expressed in the sample basis. In a common basis, the Gaussian coefficients are not iid; their covariance is MM^T, with M_{jk}=⟨φhat_j,φ_k⟩, not I. The missing term, roughly (J^{-1}‖Πhat_J−Π_J‖_HS^2)^{1/2}, is not shown to be o_P(1) under (C4), and it is not covered by Lemmas D.2–D.3. So even if the multivariate CLT and bias terms were exactly right, the displayed proof of Proposition B.1 would not imply the stated H-valued Gaussian approximation.

None of this means the theorem is false—the rotation term may well be negligible under the eigenvalue conditions, and the paper's overall strategy is plausible. But as submitted, the central result rests on at least one invalid step plus two deferred proofs. The author is honest about the omissions, which is in their favor, but the reader should not yet cite the Gaussian approximation as established.

Who should read it: functional data theorists and anyone building bootstrap inference for FPCR. It deserves a serious referee—this is exactly the kind of paper that should go to review with a request for complete proofs, not be desk-rejected. My recommendation: send it out, but the referee should be asked to verify Proposition B.1 and the omitted lemmas before acceptance.

Best,

Referee Report

4 major / 5 minor

Summary. The paper studies inference for the slope function β in the scalar-on-function regression model Y = α + ⟨β,X⟩ + ε. It defines the FPCR estimator β̂_J based on the first J sample principal components and the operator-scaled statistic T_J = √(n/J) Γ̂_J^{1/2}(β̂_J − β). The main theorem (Theorem 2.1) states that, under Conditions (C1)–(C4), (E), (B_v) with v>2, and (R_w) with w>6, the conditional Wasserstein distance between T_J and (i) σJ^{-1/2}G_J (a Gaussian element with covariance Π_J) and (ii) a residual bootstrap statistic T*_J is o_P(1). The proof decomposes T_J into a variance term plus a bias term (Eq. B.1); the latter is further decomposed into (D.1)–(D.4) and controlled via perturbation theory. Section 3 applies the result to tests of H0: β=0 using L2 and sup norms, with a simulation study.

Significance. If correct, the result overturns the long-standing no-CLT barrier for FPCR under scalar scaling (Cardot et al., 2007) and would provide a new route to asymptotic and bootstrap inference for the slope function in functional regression. The paper's operator-scaling idea is simple and novel; the bootstrap consistency is derived independently of the Gaussian approximation (Propositions B.2–B.3); the bias decomposition is explicit; and the numerical results suggest that the proposed tests improve on existing benchmarks. However, the current version does not contain complete proofs of key technical lemmas, and the Gaussian approximation proof contains a genuine rotation error. These gaps must be resolved before the central claims can be certified.

major comments (4)

[Appendix B, Proposition B.1] The proof asserts the equality W_X(√(n/J)Γ̂_J^{-1/2}U_n, σJ^{-1/2}G_J) = J^{-1/2}W_X(√n Z̄_J, σZ). This is not valid: G_J lives in span{φ_j}, whereas the first argument lives in span{φ̂_j}. In the sample basis, the coefficients of G_J are Mξ with M_{jk}=⟨φ̂_j,φ_k⟩ and ξ∼N(0,I_J), so the target covariance is MM^T, not I_J. The proof must either carry the rotation M through the multivariate CLT or bound the additional rotation term. As written, the Gaussian half of Theorem 2.1 is not established, and the stated rate O_P(n^{-1/2}J) does not follow.
[Appendix D, Lemma D.2] The proof of (D.8) states that the absolute-summability hypothesis Σ_j |⟨β,φ_j⟩|<∞ can be removed, with the argument deferred to Yeon (2026+). Since Lemma D.2 controls the bias term (D.2) and is therefore load-bearing for Theorem 2.1, the manuscript cannot rely on an unpublished companion paper for this step. Please provide a complete proof in this paper, or state the additional condition under which (D.8) holds.
[Appendix D, Lemma D.3] The perturbation bounds in (a)–(c) are asserted without proof, with the text saying 'For brevity, we omit the proof …'. These bounds are used to control terms (D.1) and (D.3) and to verify the hypothesis of Proposition B.1. Without proofs, the bias bound in Proposition B.4 and hence Theorem 2.1 are unsubstantiated. A pointer to a supplement of a different paper is not a substitute for the argument itself.
[Section 3, proof of Theorem 3.1] The proof for l=sq claims that Ψ_sq(x)=||x||^2 is Lipschitz on L^2 with constant 1; this is false since | ||x||^2 − ||y||^2 | ≤ (||x||+||y||)||x−y||, so Lemma E.2 does not apply directly. For l=sup, the proof introduces a J^{-1/4} rescaling without explanation, and the density bound used (Lemma E.6(b), constant CJ^{1/4}) does not match the target σJ^{-1/2}G_J after rescaling. The Kolmogorov-distance claims in Theorem 3.1 therefore need a rigorous proof, e.g., via a truncation argument and a correct density estimate.

minor comments (5)

[Appendix B, Proposition B.1] The symbol G_J is used both for the H-valued Gaussian and for a J-dimensional standard normal vector in the same proof, which contributes to the rotation error. Please use distinct notation for these two objects.
[Proposition A.1] The notation Γ̂_J is used for both the truncated sample covariance operator and the J×J covariance matrix of the projected regressors. Please disambiguate to avoid confusion.
[Lemma E.6] The proof of the density bound (b) is only sketched via Ball (1993) and Raič (2019). Please state precisely which theorem is applied and provide the intermediate steps, as the bound CJ^{1/4} is not immediate.
[Section 2.3] Condition (R_w) is stated as J=o(n^{1/w}) for w∈(0,∞); in Theorem 2.1 it is used with w>6. It would be helpful to note the implied range J=o(n^{1/6}) explicitly, since it is quite restrictive.
[References] The paper cites 'Yeon 2026+' and 'Choi and Park 2026+' as forthcoming. For the central Lemma D.2 this is a substantive issue (see major comment), but the citations themselves should be updated to include available preprints or DOIs when possible.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the Gaussian and bootstrap results are obtained from variance, bias, and coupling bounds; the only burden is reliance on the author's own prior/forthcoming technical lemmas, which is a verification gap rather than a circular reduction.

full rationale

I walked the derivation chain from (B.1)-(B.2) through Propositions B.1-B.4 to Theorem 2.1. The target distributions are not assumed: Lemma C.1 computes E_X[||GammaHat_J^{-1/2} U_n||^2] = sigma^2 J/n, which is what makes the J^{-1/2} scaling non-degenerate in Proposition B.1; Proposition B.3 is a coupling inequality that reduces the bootstrap component to W_X(Q,hat Q); Proposition B.2 bounds that distance using residual-error differences and Lemma C.1; Proposition B.4 handles the bias through the algebraic decomposition (D.1)-(D.4) and functional-calculus bounds. None of these steps equates Theorem 2.1 to its hypotheses, and no parameter is fitted and then called a prediction. The self-citations are real but technical: Lemma D.2 defers to Yeon (2026+) for removing sum |<beta,phi_j>|<infinity in proving (D.8), and Lemma D.3 is stated with 'For brevity, we omit the proof of Lemma D.3, as it follows from the standard argument...' and cited to Yeon et al. (2023, Lemma S3). These are auxiliary perturbation bounds that are load-bearing but are not the theorem's conclusion and are not independently assumed consequences of it; they are better viewed as omitted-proof/correctness risks. The skeptic's eigenbasis-rotation objection to the displayed scaling in Proposition B.1 is likewise a non-circular mathematical gap: it concerns whether W_X(sqrt(n/J)GammaHat_J^{-1/2} U_n, sigma J^{-1/2} G_J) equals J^{-1/2} times a coefficient-space Wasserstein distance, not a definitional circularity. Overall: no significant circularity; score 2 reflects the mild self-citation/proof-detail burden rather than any circular step.

Assumptions & free parameters 1 free parameters · 7 assumptions · 0 invented entities

The central theorem does not fit any parameters; the only input choices are technical rate conditions on J and smoothness conditions on beta. The key proof ingredient is a set of perturbation-theoretic bounds borrowed or deferred from Cardot et al. (2007), Yeon et al. (2023), and Yeon (2026+). No new particles, forces, or entities are introduced; G_J is the standard limiting Gaussian random element.

free parameters (1)

Truncation level J_n = data-driven via FVE >= 0.75 in simulations; diverging sequence in theory
J is a tuning parameter of FPCR. The theorems only require J to satisfy C4 and R_w (e.g. J = o(n^{1/6}) for w > 6); the simulations choose J by explained variance. It is not fitted to optimize the approximation, but it is a choice that affects the method's practical behavior.

assumptions (7)

domain assumption The Hilbert space H and covariance operator Gamma satisfy A1-A2: Gamma is injective and has positive distinct eigenvalues; X has finite second moment.
Section 2.1; standard for identification and for applying perturbation theory to eigencomponents.
domain assumption Conditions C1-C4: sup_j gamma_j^{-2} E<X-E[X], phi_j>^4 < infinity; gamma_j convex in j; sup_j gamma_j j log j < infinity; and J^{-1} + n^{-1} sum_{j<=J} delta_j^{-2} + n^{-1/2} sum_{j<=J} j log j = o(1).
Section 2.3; these control the resolvent/perturbation bounds for GammaHat - Gamma uniformly over the first J eigencomponents.
domain assumption Condition B_v with v > 2: J^v sum_{j>J} <beta, phi_j>^2 = o(1).
Controls the projection bias of beta onto the first J eigenfunctions; requires slope coefficients to decay faster than j^{-(v+1)/2} with v > 2.
domain assumption Condition E: E[epsilon^4 | X] = E[epsilon^4] < infinity and homoscedastic variance.
Used for residual-bootstrap consistency in Proposition B.2 and for the fourth-moment bound in Proposition B.1.
domain assumption Condition R_w with w > 6: J = o(n^{1/w}).
Restricts how fast J may grow; ensures the bias and bootstrap variance rates vanish relative to n.
domain assumption Perturbation lemmas D.1-D.3 and E.1 from Cardot et al. (2007), Yeon et al. (2023), and Yeon (2026+).
Used to control |||GammaHat_J^{1/2} - Gamma_J^{1/2}|||, |||PiHat_J - Pi_J|||, and |||GammaHat_J^{-1} - Gamma_J^{-1}|||; proofs are partly omitted or deferred, so their validity is load-bearing.
standard math Bonis (2020) multivariate CLT in Wasserstein distance and Ball (1993)/Raič (2019) Gaussian boundary estimates.
External theorems used to derive Gaussian approximation rates and the bounded density of the sup-norm statistic in Lemmas E.5-E.6.

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Cite this review

Pith. "Pith review of Gaussian and bootstrap approximations for functional principal component regression." pith.science (2026). https://pith.science/paper/3AO2UGQQ

@misc{pith2026260312518,
  author       = {Pith},
  title        = {Pith review of: Gaussian and bootstrap approximations for functional principal component regression},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3AO2UGQQ}},
  note         = {Machine review of arXiv:2603.12518}
}

read the original abstract

Asymptotic inference using functional principal component regression (FPCR) has long been considered difficult, largely because, upon any scalar scaling, the FPCR estimator fails to satisfy a central limit theorem, leading to the prevailing belief that it is unsuitable for direct statistical inference. In this paper, we revisit this traditional viewpoint by establishing a new result: upon suitable operator scaling, valid Gaussian and bootstrap approximations hold for the FPCR estimator. We apply this surprising finding to hypothesis testing for the significance of the slope function in functional regression models and demonstrate the strong numerical performance of the resulting tests. Our finding also provides a new framework for projection inference. While concise, our results yield powerful inferential tools for functional regression. We believe it paves the way for new lines of inferential methodology for more complex functional regression settings.

Figures

Figures reproduced from arXiv: 2603.12518 by the authors.

**Figure 3.1.** Empirical rejection rates of the bootstrap tests using statistics [PITH_FULL_IMAGE:figures/full_fig_p012_3_1.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

3 extracted references · 1 linked inside Pith

[2005]

Inference for function-on-function regression: central limit theorem and resid- ual bootstrap.To appear in Statistics and Its Interface, 2026+

Hyemin Yeon. Inference for function-on-function regression: central limit theorem and resid- ual bootstrap.To appear in Statistics and Its Interface, 2026+. Hyemin Yeon, Xiongtao Dai, and Daniel J. Nordman. Bootstrap inference in functional linear regression models with scalar response.Bernoulli, 29(4):2599 – 2626,

2026
[2007]

Inference for projec- tion parameters in linear regression: beyond d=o(n 1/2).arXiv preprint arXiv:2307.00795,

Woonyoung Chang, Arun Kumar Kuchibhotla, and Alessandro Rinaldo. Inference for projec- tion parameters in linear regression: beyond d=o(n 1/2).arXiv preprint arXiv:2307.00795,

arXiv
[2011]

High-dimensional Hilbert-Schmidt linear regression with Hilbert manifold variables.To appear in Annals of Statistics, 2026+

Changwon Choi and Byeong U Park. High-dimensional Hilbert-Schmidt linear regression with Hilbert manifold variables.To appear in Annals of Statistics, 2026+. Hyunphil Choi and Matthew Reimherr. A geometric approach to confidence regions and bands for functional parameters.Journal of the Royal Statistical Society Series B: Statis- tical Methodology, 80(1):239–260,

2026

Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.

[1] [2005]

Inference for function-on-function regression: central limit theorem and resid- ual bootstrap.To appear in Statistics and Its Interface, 2026+

Hyemin Yeon. Inference for function-on-function regression: central limit theorem and resid- ual bootstrap.To appear in Statistics and Its Interface, 2026+. Hyemin Yeon, Xiongtao Dai, and Daniel J. Nordman. Bootstrap inference in functional linear regression models with scalar response.Bernoulli, 29(4):2599 – 2626,

2026

[2] [2007]

Inference for projec- tion parameters in linear regression: beyond d=o(n 1/2).arXiv preprint arXiv:2307.00795,

Woonyoung Chang, Arun Kumar Kuchibhotla, and Alessandro Rinaldo. Inference for projec- tion parameters in linear regression: beyond d=o(n 1/2).arXiv preprint arXiv:2307.00795,

arXiv

[3] [2011]

High-dimensional Hilbert-Schmidt linear regression with Hilbert manifold variables.To appear in Annals of Statistics, 2026+

Changwon Choi and Byeong U Park. High-dimensional Hilbert-Schmidt linear regression with Hilbert manifold variables.To appear in Annals of Statistics, 2026+. Hyunphil Choi and Matthew Reimherr. A geometric approach to confidence regions and bands for functional parameters.Journal of the Royal Statistical Society Series B: Statis- tical Methodology, 80(1):239–260,

2026