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When missingness is independent of the data, the distributions of all finite-dimensional projections of the incomplete data determine the complete-data distribution, reducing every Hilbert-space testing problem here to projected finite-dime

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Incomplete observations in high-dimensional or functional spaces can be tested by averaging standard tests over finite-dimensional projections, provided missingness is independent of the data and every coordinate subset has a chance of being fully observed.

T0 review reviewed 2026-08-02 challenge →

load-bearing objection Theorem 1 is sound, but the paper's central testing claim overreaches: the reduction of composite goodness-of-fit to projected problems needs a projective identifiability property that is neither stated nor proved. the 3 major comments →

arxiv 2607.13209 v1 pith:PPUPWQHK submitted 2026-07-14 math.ST stat.TH

A unified approach for testing in Hilbert spaces on incomplete data

Daniel Gaigall , Philipp W\"ubbolding This is my paper
classification math.ST stat.TH MSC 62G1062G09
keywords bootstrapfunctional datagoodness-of-fithigh dimensional dataHilbert spacehypothesis testingprojectionincomplete data
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

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 establishes an equivalence that makes hypothesis testing possible when data live in a separable Hilbert space—covering high-dimensional vectors and functional data—but are only partially observed. The central theorem says two complete-data distributions are equal if and only if, for every projection onto a finite coordinate set, the distributions of the projected incomplete observations are equal, provided the missingness indicator is independent of the data and every finite coordinate set has positive probability of being fully observed. This reduces the testing problems of goodness-of-fit, symmetry, homogeneity, and independence to checking a family of finite-dimensional projected hypotheses, and motivates test statistics that average per-projection statistics with weights chosen by the statistician. The authors propose bootstrap quantiles as critical values and sketch a concrete characteristic-function-based normality test on incomplete data.

Core claim

On the paper's own terms, the discovery is Theorem 1: for a separable Hilbert space H of arbitrary dimension, with I the binary-coordinate missingness variable and ⊙ the Hadamard product (componentwise multiplication that blanks out unobserved coordinates), P^X = P^{X'} holds if and only if P^{π(I⊙X)} = P^{π(I⊙X')} holds for every finite-dimensional coordinate projection π, under the assumptions that X and I (and X' and I) are independent and that each finite projection has positive probability of being fully observed. The proof chains two facts: finite projections determine a distribution on a separable Hilbert space, and, under independence, the distribution of I⊙X determines the distribut

What carries the argument

The central object is the distributional equivalence of Theorem 1, powered by three ingredients: the family P(H) of projections onto finite-dimensional coordinate subspaces, which is known to determine a distribution on a separable Hilbert space; the missingness process I with binary coordinates and the Hadamard product I⊙X that blanks out unobserved coordinates; and the independence condition X⊥I together with the positivity condition P(π(I)=1)>0, which permits recovery of the law of X from the law of I⊙X by the identity P(X_1≤x_1,...,X_d≤x_d) = P(I_1 X_1≤x_1,..., I_d X_d≤x_d, I_1=1,...,I_d=1)/P(I_1=1,...,I_d=1). The proof chains projection determination with this conditional recovery, and

Load-bearing premise

The load-bearing premise is that missingness is independent of the data values and that every finite coordinate set has positive probability of being fully observed, since the identity that recovers the complete-data distribution from the incomplete one rests entirely on that independence and positivity.

What would settle it

Take d=2, let I be supported on {(1,0),(0,1)} only, and set X=(U,V) with U and V independent uniform variables while X'=(U,U). Then P^X≠P^{X'}, yet for every finite projection π the distributions of π(I⊙X) and π(I⊙X') coincide because no projection ever observes both coordinates jointly—showing that the positivity condition P(π(I)=1)>0 is necessary for the theorem's equivalence.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Goodness-of-fit, symmetry, homogeneity, marginal homogeneity, and independence tests on incomplete Hilbert-space data can each be written as 'for all finite projections π the projected hypothesis holds, versus there exists π for which it fails'.
  • A test statistic may be formed as T_n = Σ_π q(π)T_n(π) with arbitrary weights chosen by the statistician, and a bootstrap quantile of T_n* under the null can serve as the critical value.
  • The same incomplete-data model covers missing entries in random vectors, monotone dropout, partially observed functions, smoothed functions, and ultra-high-dimensional vectors via a dimension d_n that tends to infinity.
  • For the normality goodness-of-fit example, the characteristic function of the projected incomplete data under the null has a closed form in terms of estimated mean, covariance, and missingness probabilities, yielding a characteristic-function-based test that is available even when the dimension exceeds the sample size.

Where Pith is reading between the lines

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

  • If the equivalence is taken as a template, any consistent finite-dimensional test for each projection—not only characteristic-function-based ones—can be averaged to yield a Hilbert-space test, so the proposal is a general reduction scheme rather than one particular test.
  • The independence assumption is the practical boundary: in settings like dropout in clinical trials, where leaving the study may depend on the unobserved outcome, the recovery identity fails; a future extension would need to model the missingness mechanism or introduce imputation.
  • In the ultra-high-dimensional regime, the requirement that every finite coordinate set be observed with positive probability may be unrealistic for large coordinate sets; a possible relaxation would restrict π to coordinate sets that are actually observable, sacrificing exact equivalence for an asymptotic version.
  • Because the weighted statistic is an average over randomly generated projections, variance reduction and replication-sparing techniques from Monte-Carlo bootstrap testing could apply directly to make the procedure computationally feasible.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The manuscript proposes a projection-based framework for hypothesis testing on incomplete observations with values in a separable Hilbert space. Incompleteness is encoded by a Bernoulli-valued coordinate indicator I and observed value I⊙X. The main mathematical result, Theorem 1 in Section 2.4, asserts that for two random elements X and X′ with I independent of both, and under positivity of all finite-dimensional complete-case probabilities, P^X = P^{X′} if and only if P^{π(I⊙X)} = P^{π(I⊙X′)} for every finite-dimensional projection π. On this basis, Section 2.5 claims that all listed testing problems — goodness-of-fit, symmetry, homogeneity, independence — can be expressed equivalently as "∀π H0(π) vs. ∃π H1(π)", and proposes tests of the form T_n = Σ_π q(π) T_n(π) with bootstrap critical values. Section 2.6 sketches the example of goodness-of-fit for normality via a BHEP-type statistic, referring to a companion paper for details. Section 3 states that the distribution theory of the proposed statistics requires further investigation.

Significance. If the central reduction were valid, the paper would provide a clean and broadly applicable identification statement: under MCAR-type missingness and positivity, finite-dimensional projected incomplete-data laws determine the full law, so that nonparametric hypotheses can be attacked by aggregating projected tests. The proof of Theorem 1 is elementary and correct under the stated assumptions, and the explicit identification formula is a useful contribution. Credit is due for identifying the key independence and positivity conditions and for embedding several missingness patterns in one framework. However, the leap from Theorem 1 to composite parametric goodness-of-fit is not justified, the only concrete example is delegated to a companion paper, and no asymptotic properties of the proposed test statistics are established. The paper is therefore better viewed as a programmatic contribution than as a completed testing methodology, and its advertised reduction is not yet a theorem.

major comments (3)
  1. [Section 2.5 and Example 3] The central reduction for goodness-of-fit is not proved. Theorem 1 is an equivalence between two fixed laws: P^X=P^{X'} ⇔ ∀π P^{π(I⊙X)}=P^{π(I⊙X')}. For the composite null H0: P^X ∈ {P^X_ϑ; ϑ∈Θ}, the reverse direction requires a projective identifiability/closure property: if for every π there exists some ϑ_π with P^{π(I⊙X)} = P^{π(I⊙X)}_{ϑ_π}, then there exists a single ϑ with P^X = P^X_ϑ. This does not follow from Theorem 1 because ϑ_π may vary with π. Example 3 simply asserts the equivalence, and Section 2.6 does not repair the gap: the claim there that φ_π = ϕ_π for all π iff H0 is stated without proof and is delegated to Gaigall and Wübbolding (2026). This is a load-bearing gap for the paper's main claim that all listed testing problems reduce to families of projected tests.
  2. [Section 2.4 and Example 2] The theorem's two key assumptions — I ⊥ X and P(π(I)=1)>0 for every finite π — are not satisfied by some of the motivating applications. In the monotone dropout model of Example 2b, D_i is specified as iid but its independence from X_i is not stated; informative dropout is precisely a concern in the cited depression-trial setting and, if D_i depends on unobserved coordinates of X_i, the identification formula in the proof of Theorem 1 collapses. In the ultra-high-dimensional regime of Example 2d, positivity fails for any projection involving coordinates beyond d_n, since P(I_n(j)=1)=0 for j>d_n. If the intended regime is d_n→∞ with positivity only on the first d_n coordinates, this needs to be stated explicitly, and the asymptotic consequences need to be developed.
  3. [Section 2.6] The normality example is not fully coherent as written. The family P^{π(I⊙X)} is described as "corresponding families of k-dimensional normal distributions", but π(I⊙X) has zero coordinates with positive probability whenever some components of I are zero; its distribution under H0 is a mixture over missingness patterns, as the displayed formula for ϕ_π shows. The equivalence "φ_π = ϕ_π for all π if and only if H0" is the key step for the example, but no proof is provided here. Since this is the only concrete illustration of the general idea, the example should either be proved within the paper or explicitly labeled as a conjecture/sketch with the proof referring to a verifiable source.
minor comments (5)
  1. [Throughout] The notation P^{π(I⊙X)} is used both for the distribution of π(I⊙X) and, in Example 3, for the parametric family {P^{π(I⊙X)}_ϑ; ϑ∈Θ}. This ambiguity is confusing; use different notation, e.g. P_ϑ^π for the projected parametric family.
  2. [Section 2.4] The proof of Theorem 1 relies on the fact that finite-dimensional projections determine distributions on H, referring to "similar arguments as in Ditzhaus and Gaigall (2018)". This is standard, but since it is the core technical step of the theorem, the paper should either give a self-contained argument or a precise statement with a full reference.
  3. [Section 2.5] If T_n is a weighted average with weight function q that is not supported on all finite-dimensional projections, a test based on T_n may have no power against alternatives for which H1(π) holds only on projections with q(π)=0. If consistency is intended, the support condition on Q/q should be stated.
  4. [Section 2.6] The estimators μ̂_n, Σ̂_n, and p̂_n(a) are introduced without any concrete definition or regularity conditions. At least one explicit estimator (or a reference to the companion paper with the relevant conditions) should be given, since the test statistic depends on them.
  5. [Section 3 and Section 2.6] Typos: "Smilar" in Section 3 should be "Similar"; "scetch" in Section 2.6 should be "sketch".

Circularity Check

0 steps flagged

No significant circularity: Theorem 1 is proved in-text; the Section 2.5 composite-family reduction is an unproved generalization, not a circular step.

full rationale

The derivation chain is not circular. Theorem 1 is argued in-text: the projection-uniqueness fact is stated explicitly ('we obtain that the projections in P(H) determines the distribution P^X of X uniquely') and the incomplete-data identification formula is displayed in the proof ('Analogously to Gaigall and Wübbolding (2026), it is P(X(1)≤x_1,...,X(d)≤x_d)=...'), so the core equivalence P^X=P^{X'} ⇔ P^{I⊙X}=P^{I⊙X'} is not an assumed input. The BHEP illustration in Section 2.6 relies on the companion paper for the concrete test but also derives the characteristic-function identity from Theorem 1. The main caveat is non-circular: Section 2.5/Example 3 asserts that Theorem 1 reduces composite goodness-of-fit to '∀π∈P(H):H0(π) vs. ∃π∈P(H):H1(π)', but Theorem 1 only equates fixed laws; for a general parametric family the reverse direction requires a projective identifiability/closure property that is not stated or proved, and ϑ_π may vary with π. This is a logical gap/correctness risk, not a circularity, and the paper itself concedes that the test statistics' development 'requires independent investigations.' Self-citations are numerous and structural, but the load-bearing identification is displayed in the text rather than imported as an unverified black box, and the projected-family construction does not define the target conclusion into existence. I therefore find no significant circularity; the score of 1 reflects only the heavy structural reliance on the authors' prior framework, not a circular step.

Axiom & Free-Parameter Ledger

3 free parameters · 5 axioms · 0 invented entities

The central claims rest on stated assumptions: i.i.d. copies of (I, I⊙X), independence of I from X (MCAR), and positivity of complete-case probabilities for every finite coordinate set. Theorem 1 invokes standard facts about projections determining distributions. The framework leaves unspecified the estimator choices (μ̂_n, Σ̂_n, p̂_n), the projection weight q, and the projection-sampling distributions ν and η; the asymptotic validity of the resulting tests is deferred to future work. No genuinely new entities are postulated; the mask I is a standard missingness device, not an invented entity.

free parameters (3)
  • Weight function q / probability measure Q on P(H)
    Section 2.5: Q (or q) is 'chosen by the statistician'; the distribution of T_n and hence the critical values depend on this unspecified choice.
  • Projection-sampling distributions ν and η
    Section 2.5 Monte-Carlo step: realizations of π are generated from arbitrary discrete distributions ν on J and η on {1,...,k}; test power depends on these choices.
  • Estimators μ̂_n, Σ̂_n, p̂_n(a)
    Section 2.6: 'Indeed, several choices for the estimators are possible, see Gaigall and Wübbolding (2026).' The null distribution of the BHEP statistic depends on these estimator choices, which are delegated to the companion paper.
axioms (5)
  • standard math Finite-rank coordinate projections determine the distribution on a separable Hilbert space; the characteristic functional determines the distribution
    Invoked in the proof of Theorem 1 (Section 2.4), via Ditzhaus and Gaigall (2018) and Laha and Rohatgi (1979).
  • domain assumption Missingness indicator I is independent of X (and of X')
    Stated in Theorem 1 (Section 2.4). The identification identity in the proof factorizes P(I⊙X ∈ ·, I=1) = P(I=1) P(X ∈ ·); this is an MCAR assumption that fails under mechanism-dependent missingness such as informative dropout (Example 2b).
  • domain assumption Positivity: P(π(I)=1) > 0 for every π ∈ P(H)
    Assumed in Theorem 1 (Section 2.4). The unconditional distribution of X is recovered only from observations where all relevant coordinates are observed; strained in the ultra-high-dimensional regime d_n→∞ (Example 2d).
  • domain assumption Observations (I_i, I_i⊙X_i) are i.i.d. copies of (I, I⊙X)
    Section 2.2. The paper's 'specific structure' that licenses LLN/CLT arguments is assumed, not derived.
  • ad hoc to paper For parametric goodness-of-fit, ∀π: P^{π(I⊙X)} ∈ P^{π(I⊙X)} implies P^X ∈ P^X
    Section 2.5 asserts the general parametric reduction as a deduction from Theorem 1, but Theorem 1 only covers equality of two fixed laws. The ∀π/∃ϑ interchange requires a family-specific closure property that is neither stated nor proved; it happens to hold for the normality example (Section 2.6).

reviewed 2026-08-02 · how reviews work

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Cite this review

Pith. "Pith review of A unified approach for testing in Hilbert spaces on incomplete data." pith.science (2026). https://pith.science/paper/PPUPWQHK

@misc{pith2026260713209,
  author       = {Pith},
  title        = {Pith review of: A unified approach for testing in Hilbert spaces on incomplete data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PPUPWQHK}},
  note         = {Machine review of arXiv:2607.13209}
}
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read the original abstract

We consider statistical testing on the basis of incomplete observations with values in a separable Hilbert space, where the dimension is possibly large or even infinite. The general Hilbert space setting allows various data types as they arise in modern applications, in particular high dimensional and functional data. Possible Hilbert space testing problems are goodness-of-fit, symmetry, homogeneity and independence. We present an approach for modeling incomplete data that covers several problems in practice, e.g., ultra high dimensional random vectors with missing entries or partially observed stochastic processes. We identify a specific structure (independent and identically distributed) in the incomplete data that enables the analysis of statistical procedures with the help of suitable mathematical results (e.g., laws of large numbers and central limit theorems). Additionally, a general and novel concept for testing different hypotheses in this situation is suggested and sketched for the example of testing goodness-of-fit for normality.

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This paper was first reviewed by deepseek-v4-flash on August 2, 2026.