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One scalar calibrates shrinkage covariance monitors across elliptical returns

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2026-07-08 07:50 UTC pith:VDHEHBYW

load-bearing objection The kurtosis-immunity converse and the null-transfer identity are genuinely new; the high-dimensional coverage gap is real but honestly reported. the 3 major comments →

arxiv 2607.06373 v1 pith:VDHEHBYW submitted 2026-07-07 stat.ME math.STq-fin.STstat.COstat.TH

Error Propagation in Spectral Functionals of Shrinkage Covariance Estimators: Perturbation Bounds and Calibrated Inference

Ahmad Koman This is my paper
classification stat.ME math.STq-fin.STstat.COstat.TH
keywords functionalsshrinkagewidehatcovarianceeigenspacescalarabsorptioncalibrated
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.

This paper asks a question that practitioners face daily: when the dominant eigenspace or the absorption ratio of a rolling covariance estimate moves, is that movement genuine structural change or just estimation noise? The author builds a calibration pipeline around the finding that scale-invariant spectral functionals (the absorption ratio, the leading-eigenvalue share, and the top-K eigenspace projector) are immune to elliptical kurtosis at first order, and that this immunity holds only for scale-invariant functionals. The mechanism is Euler's relation for degree-zero homogeneous functions: the common-shock fourth-moment term that plagues level functionals like total variance vanishes identically for ratio functionals, so a single estimated scalar (the kurtosis parameter) calibrates the projector null, the absorption-ratio intervals, and the leading-share intervals simultaneously across the entire elliptical family. For the eigenspace monitor, the author derives a first-order null law for projector movement between overlapping windows that share most of their data, shows it transfers without modification to rotation-equivariant shrinkage estimators (because such estimators preserve eigenvectors), and supplies a parametric bootstrap for the cases the analytic null misses. A companion power analysis gives closed-form expressions for the smallest rotation a two-window monitor can detect and the number of fresh observations required, exposing an onset power ceiling where quarter-turn rotations are undetectable at intermediate overlap. In high dimensions, the author identifies an explicit wedge: sample eigenvalues bias the absorption ratio upward via Marchenko-Pastur spreading, Frobenius-optimal shrinkage biases it downward by a comparable margin, and only a trace-preserving spike-debiased estimator removes the bias.

Core claim

The central object is the scale-invariance property of spectral functionals under the elliptical distribution family. The author proves that first-order kurtosis immunity (where the asymptotic variance scales by exactly (1+kappa) and no higher) holds for scale-invariant functionals and only for them, with the converse established via an Euler-relation argument along ray-connected open sets. This means the top-K projector, the absorption ratio, and the leading-eigenvalue share all shed the common-volatility-shock term that dominates level functionals under heavy tails, so one estimated kurtosis scalar calibrates all three. The second central result is the transfer principle for rotation-equiv

What carries the argument

Davis-Kahan sin-theta perturbation bound capped at maximum projector distance; first-order null law for overlapping-window projector movement driven by fresh observations only; Euler relation for degree-zero homogeneous functions proving kurtosis immunity; spiked-model bias inversion (Marchenko-Pastur spike map) for trace-preserving absorption-ratio debiasing; median-shift power approximation yielding closed-form detectability frontier

Load-bearing premise

The analytic calibration rests on i.i.d. elliptical sampling, which real return panels systematically violate through volatility clustering (time-varying scales within the window) and skewness; under log-normal innovations the kurtosis correction breaks entirely, with coverage falling to 48 percent against a 95 percent target.

What would settle it

Find a scale-invariant spectral functional of the covariance whose first-order asymptotic variance under i.i.d. elliptical sampling does not factor as (1+kappa) times its Gaussian variance, or find a non-scale-invariant functional that does enjoy such factoring; either would break the immunity-and-converse characterization of Proposition 8.

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

If this is right

  • Practitioners using shrinkage-cleaned covariance matrices for eigenspace monitoring need not re-derive null distributions for each shrinkage estimator: rotation-equivariant maps (linear shrinkage, QIS) preserve the eigenvectors and thus the sample-covariance null applies directly, halving bootstrap cost.
  • The cleaning-debiasing wedge means that computing the absorption ratio from the same cleaned matrix used for the projector monitor introduces a known bias of opposite sign to the sample bias, so risk monitors should use spike-debiased eigenvalues for ratio functionals and cleaned eigenvalues for projectors separately.
  • The detectability frontier formula gives portfolio managers a concrete design rule: the smallest detectable eigenspace rotation scales as 1/sqrt(n_post), so halving the detectable angle requires four times as many post-break observations, and at onset a quarter-turn rotation can be entirely invisible.
  • The exact window-scale pivotality (Proposition 9) explains why spectral monitors are blind to pure volatility changes and react only to correlation-structure changes, a property that holds exactly in finite samples under a single window-wide scale.

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 / 6 minor

Summary. This paper develops calibration tools for spectral monitoring of rolling covariance estimates under shrinkage. It contributes: (1) a distribution-free Davis–Kahan band for sequential top-K projector movement, capped at the maximum projector distance; (2) a first-order null law for projector movement between overlapping windows that transfers to rotation-equivariant shrinkage estimators; (3) a local power analysis yielding a detectability frontier for rotation alternatives; (4) a kurtosis-immunity principle showing that scale-invariant spectral functionals depend on the elliptical family only through a single scalar κ, with a converse characterization; (5) confidence intervals for the absorption ratio and leading-eigenvalue share; and (6) a trace-preserving spike-debiased absorption-ratio estimator for high dimensions. The results are verified by simulation under known population covariance, with an equity-panel illustration.

Significance. The paper addresses a practically important problem—distinguishing estimation noise from structural change in rolling covariance monitors—and provides a coherent calibration pipeline spanning both subspace movement and scalar spectral functionals. The kurtosis-immunity characterization (Proposition 8) with its converse is a clean theoretical contribution, and the cleaning–debiasing wedge (Proposition 11) gives a sharp diagnostic for a real pitfall in high-dimensional absorption-ratio inference. The transfer of the sample-covariance null to rotation-equivariant shrinkage (§4.1) is a useful simplification. Simulations are well-designed with known population covariance, and the detectability frontier (Corollary 3) provides a falsifiable design rule. The equity-panel appendix is appropriately scoped as diagnostic rather than confirmatory.

major comments (3)
  1. Abstract and §5.2–§5.3: The abstract states that 'one estimated scalar calibrates the absorption-ratio intervals across the elliptical family,' but Table 7 shows this claim fails in the regime where the method is deployed. At N=115, M=252 (N/M≈0.46) under elliptical t₅ with κ_true=2, the delta-method AR interval achieves only 75.0% coverage versus 95% nominal. The 91.2% coverage reported for the HD-debiased version with radial-MLE κ̂ is rescued by the conservative bias of κ̂ (acknowledged in Remark 4: 'conservative for extreme tails (t₅)'), not by the (1+κ) correction itself. Proposition 8 is a fixed-N, M→∞ result, but the simulations and equity panel operate in the proportional regime. The abstract's claim should be qualified by dimension regime, or the paper should clarify that the practical contribution is the fixed-N CLT while the high-dimensional case remains open (as §5.4 partially
  2. §5.4, Proposition 11(iii): The CLT for the debiased absorption-ratio estimator is stated for the relative error √M(AR̂_deb/AR−1)→N(0,V_rel), but V_rel is left unspecified ('of order one'). Since this is the only inferential guarantee for the debiased estimator in the proportional regime—the paper's central practical setting—the variance formula should be given explicitly, or the paper should state clearly that the CLT is a consistency result without a usable variance for interval construction. Table 7 reports intervals for the debiased estimator, so the reader needs to know what variance is being plugged in.
  3. §4.2, Proposition 2: The first-order null is derived under Gaussian sampling, with the elliptical extension in Remark 3 stated as a variance scaling by (1+κ). However, the proof in Appendix A.3 computes the variance of the increment entries g_ij under Gaussian y_ik∼N(0,λ_i), and the elliptical extension is asserted via the fourth-moment tensor (16) without a separate derivation. Given that the projector null is load-bearing for the monitoring pipeline, the elliptical extension should be verified explicitly—particularly whether the shared-block cancellation that drives the overlapping-window structure survives under elliptical sampling, or whether additional cross-terms appear at order s/M².
minor comments (6)
  1. Table 3 vs. Table 4: The |AR_K err| values for the Sample estimator differ (0.0877 vs. 0.0302) despite identical N, M, and seed. If K differs between the two tables, this should be noted; if not, the discrepancy needs explanation.
  2. §3.1, Theorem 1: The constant c_K = 2√(2K) is described in Remark 2, but the per-date term T_{K,s} in (4) uses 2√2·min(√K·η_t, ‖E_t‖_F)/Δ_{K,t}. The relationship between the min(·) form and the simplified bound should be clarified for readers who may use the simplified version.
  3. Figure 1 caption: 'C_A = Σ + εI' and 'C_B = (1+b)Σ' are referenced but the figure labels show 'C_A = +I' and 'C_B = (1+b)', missing the Σ symbol.
  4. §6.4: The replication counts (150–200 per cell) are modest for size calibration at the 5% level; a note on Monte Carlo error for the reported flag rates would help the reader assess precision.
  5. Appendix B, Table 12: The moment-based κ̂ (median ≈1.5) and radial-MLE κ̂ (≈0.2) disagree substantially. While §B.2 attributes this to within-window volatility clustering, the magnitude of the discrepancy (7.5×) is large enough that a brief discussion of which is more reliable for panel data would be valuable.
  6. References: Paindaveine et al. (2026) and Lin and Pan (2026) are cited but appear to be forthcoming; please verify final publication details.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for a careful and constructive report. All three major comments identify genuine gaps between claims and evidence, and we address each below. In brief: (1) the abstract overstates the kurtosis-calibration claim for the proportional regime and will be qualified; (2) the variance V_rel in Proposition 11(iii) can and will be given explicitly; (3) the elliptical extension of the overlapping-window null is currently asserted via Corollary 4 rather than derived in the proof of Proposition 2, and we will add the explicit computation.

read point-by-point responses

  1. Referee: Abstract and §5.2–§5.3: The abstract states that 'one estimated scalar calibrates the absorption-ratio intervals across the elliptical family,' but Table 7 shows this claim fails in the regime where the method is deployed. At N=115, M=252 (N/M≈0.46) under elliptical t₅ with κ_true=2, the delta-method AR interval achieves only 75.0% coverage versus 95% nominal. The 91.2% coverage reported for the HD-debiased version with radial-MLE κ̂ is rescued by the conservative bias of κ̂, not by the (1+κ) correction itself. Proposition 8 is a fixed-N, M→∞ result, but the simulations and equity panel operate in the proportional regime.

    Authors: The referee is correct on all counts. Proposition 8 (kurtosis immunity) is a fixed-N, M→∞ result, and the abstract's claim that 'one estimated scalar calibrates the absorption-ratio intervals across the elliptical family' is stated without this qualification. Table 7 makes the gap explicit: at N=115, M=252 under t₅, the delta-method interval with κ_true achieves only 75.0% coverage—well below the 95% nominal—because Marchenko–Pastur spreading introduces a bias that the (1+κ) correction alone cannot absorb. The 91.2% coverage of the HD-debiased interval with radial-MLE κ̂ is indeed partly attributable to the conservative upward bias of κ̂ under t₅ (Remark 4: 'conservative for extreme tails'), not solely to the debiasing or the kurtosis correction. The paper acknowledges these limitations in §5.4 and the Discussion ('Fully calibrated absorption-ratio inference under elliptical heavy tails in high dimensions remains open'), but the abstract does not reflect them. We will revise the abstract to qualify the calibration claim by dimension regime: the fixed-N CLT (Proposition 10) provides valid kurtosis-calibrated intervals across the elliptical family, while in the proportional regime the (1+κ) correction is necessary but not sufficient, and fully calibrated AR inference under elliptical heavy tails at N/M ≈ O(1) remains open. We will also add a sentence in §5.3 noting that the 91.2% coverage in Table 7 benefits from the conservative direction of the radial-MLE κ̂ under t₅, so the reader does not infer that the (1+κ) factor alone closes the gap. revision: yes

  2. Referee: §5.4, Proposition 11(iii): The CLT for the debiased absorption-ratio estimator is stated for the relative error √M(AR̂_deb/AR−1)→N(0,V_rel), but V_rel is left unspecified ('of order one'). Since this is the only inferential guarantee for the debiased estimator in the proportional regime—the paper's central practical setting—the variance formula should be given explicitly, or the paper should state clearly that the CLT is a consistency result without a usable variance for interval construction. Table 7 reports intervals for the debiased estimator, so the reader needs to know what variance is being plugged in.

    Authors: The referee is right that V_rel is left unspecified and that this is unsatisfactory given that Table 7 reports intervals for the debiased estimator. The variance can be written explicitly. By Bai and Yao (2008), the spike eigenvalue fluctuations √M(λ̂_i − ψ(λ_i)) are asymptotically independent Gaussians with variance σ²_{λ,i} = 2cσ⁴λ_i²/(λ_i − σ²)² (for Gaussian observations in the spiked model). The delta method through ψ⁻¹, whose derivative is ψ'(λ) = 1 − cσ⁴/(λ − σ²)², gives V_rel = (1/S_K²) Σ_{i≤K} σ²_{λ,i} / [ψ'(λ_i)]². We will add this formula to Proposition 11(iii) and specify in §5.4 that the intervals in Table 7 plug in the empirical estimates ĉ, σ̂², λ̂_i, and the radial-MLE κ̂ (which enters through the elliptical extension of the Bai–Yao variance, replacing σ²_{λ,i} by (1+κ̂)σ²_{λ,i} for the off-diagonal fourth-moment contribution). We will also note that this variance formula is derived under Gaussian sampling and that its validity under elliptical heavy tails in the proportional regime is not established—the same caveat that applies to the coverage numbers themselves. revision: yes

  3. Referee: §4.2, Proposition 2: The first-order null is derived under Gaussian sampling, with the elliptical extension in Remark 3 stated as a variance scaling by (1+κ). However, the proof in Appendix A.3 computes the variance of the increment entries g_ij under Gaussian y_ik∼N(0,λ_i), and the elliptical extension is asserted via the fourth-moment tensor (16) without a separate derivation. Given that the projector null is load-bearing for the monitoring pipeline, the elliptical extension should be verified explicitly—particularly whether the shared-block cancellation that drives the overlapping-window structure survives under elliptical sampling, or whether additional cross-terms appear at order s/M².

    Authors: The referee raises a valid concern. The shared-block cancellation itself is exact and distribution-free: the M−s shared observations enter both window covariances identically and cancel in the difference S_t − S_{t−1} regardless of the sampling distribution. What requires verification is the variance structure of the increment entries g_{ij} under elliptical sampling, including cross-pair covariances. The argument is present in outline in Corollary 4 but is not spelled out in the proof of Proposition 2, and we agree it should be. The computation is as follows. In the population eigenbasis, the elliptical fourth-moment tensor (16) gives Cov(S_{ij}, S_{kl}) = (1+κ)(λ_iλ_k δ_{ik}δ_{jl} + λ_iλ_l δ_{il}δ_{jk}) + κλ_iλ_k δ_{ij}δ_{kl}. For the off-diagonal entries g_{ij} with i≤K<j (hence i≠j), the common-shock term κλ_iλ_k δ_{ij}δ_{kl} vanishes because δ_{ij}=0. The remaining terms are exactly (1+κ) times the Gaussian fourth-moment structure, so both the marginal variances Var(g_{ij}) = 2s(1+κ)λ_iλ_j/M² and the cross-pair covariances Cov(g_{ij}, g_{kl}) scale by (1+κ) relative to the Gaussian case. Since the first-order law (8) is a function of the g_{ij} alone, its entire distribution scales by √(1+κ), with no additional cross-terms at order s/M². This is the content of Corollary 4, but it is currently stated as a consequence of Proposition 8 rather than verified directly in the proof of Proposition 2. We will add this computation to Appendix A.3 as a separate lemma or remark, making the elliptical extension of the null a derived result rather than an assertion. One caveat we will note: the argument relies on the i.i.d. structure across observations (independence of the radial component across k), which is part of the elliptical model but would fail under volatility聚类; this残 revision: no

Circularity Check

0 steps flagged

No circularity found: derivations are parameter-free and grounded in external mathematical results

full rationale

The paper's central results are genuine derivations from established external mathematics, not circular reductions to fitted inputs. Proposition 8's variance formula (17) follows from the elliptical fourth-moment tensor (16), attributed to Tyler 1981, Browne 1984, and Muirhead 1982, combined with Euler's relation for degree-zero homogeneity. The immunity direction (ii) is a direct consequence: scale invariance forces ⟨G,Σ⟩=0, eliminating the common-shock term. The converse (iii) is the paper's own contribution, proved in Appendix A.8 by integrating the gradient condition along rays—no self-citation is involved. Proposition 10 is explicitly 'immediate from Proposition 8(ii) with the gradients of Lemma 6,' a straightforward instantiation. Proposition 11's wedge formulas derive from spiked-model asymptotics (Paul 2007; Baik and Silverstein 2006; Benaych-Georges and Nadakuditi 2011), all external. The α-scaling (Remark 1) is transparently presented as a calibration step ('calibration selects the smallest α⋆ on the grid at which the simulated flag rate under the null stays at or below 5%'), not as a first-principles prediction. The κ̂ estimator (Remark 4) is a plug-in for the theoretically derived (1+κ) factor; the prediction is the form of the correction, not the value of κ. The paper acknowledges limitations honestly (§5.4, Discussion: 'fully calibrated AR inference for elliptical heavy tails in high dimensions remains an open problem'). No step in the derivation chain reduces to its own inputs by construction.

Axiom & Free-Parameter Ledger

5 free parameters · 7 axioms · 0 invented entities

No new particles, forces, dimensions, or postulated entities. The 'trace-preserving spike-debiased estimator' (§5.4) is a new estimator construction, not a new entity—it is defined by inverting the known spiked-model bias map ψ(λ) and redistributing the spike inflation to the bulk. The axiom ledger is dominated by the elliptical and rotation-equivariance assumptions, both of which are standard domain assumptions with documented failure modes in the paper.

free parameters (5)
  • α (band scale) = 0.75 (grid-calibrated on simulated null)
    Smallest grid value whose simulated calm-period flag rate stays ≤5% (Remark 1, §6.2). Operational choice, not a theoretical constant.
  • κ (kurtosis parameter) = estimated per-window via radial-MLE
    Estimated from data; not a free parameter of the theory but an input to the calibrated intervals. Capped at 0 for ν>100 (Remark 4).
  • K (monitoring dimension) = selected by dominant eigengap, capped at K_max
    Data-driven selection (Proposition 1); K_max is a user configuration.
  • M (window length) = 252 (daily, ~1 year)
    User configuration, not fitted.
  • B (bootstrap draws) = 60-99
    Computational budget choice.
axioms (7)
  • domain assumption Returns are i.i.d. elliptical with finite fourth moments for the kurtosis-immunity theory (§5.2, Proposition 8)
    The (1+κ) correction and the CLT of Proposition 10 depend on the elliptical fourth-moment tensor (16). Table 8 shows this fails for skewed data.
  • domain assumption The covariance estimator A is rotation-equivariant (§2)
    Required for the null-transfer result of §4.1. QIS and LW satisfy it; MP-clip does not (eigenvalue ties).
  • domain assumption Eigengap Δ_{K,t} > 0 (positive population gap at the monitoring cut)
    Required for Davis-Kahan (Lemma 2), the per-date deviation (Lemma 4), and the monitoring band (Theorem 1).
  • domain assumption η_s < Δ/4 for the second-order remainder bound (Proposition 3)
    Controls the Neumann series convergence. The paper states this 'typically fails' at panel N/M (§4.2), which is when the bootstrap is needed.
  • domain assumption Spiked model: Σ = σ²I + Σ_spikes, fixed distinct spikes above bulk edge (Proposition 11)
    Required for the cleaning-debiasing wedge asymptotics. Standard in RMT.
  • standard math Standard matrix perturbation theory (Davis-Kahan, Weyl, Kato expansion)
    Background results cited correctly (Yu et al. 2015; Stewart-Sun 1990).
  • standard math Spiked-model asymptotics (Paul 2007; Baik-Silverstein 2006; Benaych-Georges-Nadakuditi 2011)
    External results used for Proposition 11 wedge formulas.

pith-pipeline@v1.1.0-glm · 42477 in / 3681 out tokens · 342700 ms · 2026-07-08T07:50:51.419588+00:00 · methodology

0 comments
read the original abstract

Rolling covariance estimates feed two objects that are routinely treated as market structure. The first is the dominant eigenspace, monitored through the projector movement $\widehat D_{K,t}=\|\widehat P_{K,t}-\widehat P_{K,t-1}\|_F$; the second comprises scalar spectral functionals such as the absorption ratio and the leading-eigenvalue share. Both fluctuate under estimation noise, and shrinkage changes the law of that noise, so reading their movements as structural change requires calibration. For the eigenspace, we derive a first-order null law for $\widehat D_{K,t}$ between overlapping windows that share most of their data and show that it transfers without change to rotation-equivariant shrinkage estimators. A distribution-free Davis-Kahan band gauges whether the eigenspace is identified, an estimator-aware bootstrap provides the calibrated test, and a companion power analysis gives an approximate design rule for the smallest detectable rotation. For the scalar functionals, we show that first-order immunity to elliptical kurtosis holds for scale-invariant functionals and only for them, so that one estimated scalar calibrates the projector null and the absorption-ratio and leading-share intervals across the elliptical family. In high dimensions, where shrinkage cleaning biases the absorption ratio, we give a trace-preserving spike-debiased estimator that removes the bias. The results are verified by simulation under a known population covariance; an equity-panel appendix shows the procedures as diagnostics when the population is unknown.

Figures

Figures reproduced from arXiv: 2607.06373 by Ahmad Koman.

Figure 1
Figure 1. Figure 1: Diagonal example (N=10, K=3), where CA = Σ + εI has smaller Frobenius distance than scaled CB = (1 + b)Σ but larger |ARK| error (Proposition 6). The same separation holds for the other functionals. The proof of Proposition 6 uses only scale invariance and ϕ ′ (0) ̸= 0, so it applies verbatim to f1 = λ1/T and to the spectral entropy of pi = λi/ P j λj . Proposition 7 (Functional error propagation). Fix posi… view at source ↗
Figure 2
Figure 2. Figure 2: Simulated flag rate vs. α (known Σt , population τ ∗ K,t) [PITH_FULL_IMAGE:figures/full_fig_p021_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Simulated mean functional error by estimator (known Σ, [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Simulated eigenmonitor volatility by gap [PITH_FULL_IMAGE:figures/full_fig_p022_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Power of the analytic test, predicted (linear and mixture-exact first-order laws, pop [PITH_FULL_IMAGE:figures/full_fig_p024_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: First-order null of the projector movement [PITH_FULL_IMAGE:figures/full_fig_p024_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Cleaning–debiasing wedge across concentration [PITH_FULL_IMAGE:figures/full_fig_p027_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Daily projector movement DbK,t on the panel against the elliptical-scaled analytic threshold (moment-based κbt , §5.2), with the 2008 financial crisis and the 2020 COVID window shaded. Observed movement stays below the threshold in calm periods and exceeds it in clusters during market stress; the threshold widens, and can saturate, when identification is weak, so a few weak-identification dates fall above … view at source ↗

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Reference graph

Works this paper leans on

42 extracted references · 42 canonical work pages

  1. [1]

    Ahn and Alex R

    Seung C. Ahn and Alex R. Horenstein. Eigenvalue adjusted estimated correlation matrices. Econometric Reviews, 32 0 (5-6): 0 691--712, 2013

    work page 2013

  2. [2]

    Eigenvector dynamics: general theory and some applications

    Romain Allez and Jean-Philippe Bouchaud. Eigenvector dynamics: general theory and some applications. Physical Review E, 86: 0 046202, 2012

    work page 2012

  3. [3]

    T. W. Anderson. Asymptotic theory for principal component analysis. Annals of Mathematical Statistics, 34 0 (1): 0 122--148, 1963

    work page 1963

  4. [4]

    Z. D. Bai and Y. Q. Yin. Limit of the smallest eigenvalue of a large dimensional sample covariance matrix. Annals of Probability, 21 0 (3): 0 1275--1294, 1993

    work page 1993

  5. [5]

    Central limit theorems for eigenvalues in a spiked population model

    Zhidong Bai and Jianfeng Yao. Central limit theorems for eigenvalues in a spiked population model. Annales de l'Institut Henri Poincar\'e, Probabilit\'es et Statistiques, 44 0 (3): 0 447--474, 2008

    work page 2008

  6. [6]

    Silverstein

    Jinho Baik and Jack W. Silverstein. Eigenvalues of large sample covariance matrices of spiked population models. Journal of Multivariate Analysis, 97 0 (6): 0 1382--1408, 2006

    work page 2006

  7. [7]

    The eigenvalues and eigenvectors of finite, low rank perturbations of large random matrices

    Florent Benaych-Georges and Raj Rao Nadakuditi. The eigenvalues and eigenvectors of finite, low rank perturbations of large random matrices. Advances in Mathematics, 227 0 (1): 0 494--521, 2011

    work page 2011

  8. [8]

    Matrix Analysis

    Rajendra Bhatia. Matrix Analysis. Springer, New York, 1997

    work page 1997

  9. [9]

    Michael W. Browne. Asymptotically distribution-free methods for the analysis of covariance structures. British Journal of Mathematical and Statistical Psychology, 37 0 (1): 0 62--83, 1984

    work page 1984

  10. [10]

    Cleaning large correlation matrices: tools from random matrix theory

    Joel Bun, Jean-Philippe Bouchaud, and Marc Potters. Cleaning large correlation matrices: tools from random matrix theory. Physics Reports, 666: 0 1--109, 2017

    work page 2017

  11. [11]

    Overlaps between eigenvectors of correlated random matrices

    Jo \"e l Bun, Jean-Philippe Bouchaud, and Marc Potters. Overlaps between eigenvectors of correlated random matrices. Physical Review E, 98: 0 052145, 2018

    work page 2018

  12. [12]

    Hero, and Yonina C

    Yilun Chen, Ami Wiesel, Alfred O. Hero, and Yonina C. Eldar. Shrinkage algorithms for MMSE covariance estimation. IEEE Transactions on Signal Processing, 58 0 (10): 0 5016--5029, 2010

    work page 2010

  13. [13]

    The rotation of eigenvectors by a perturbation, III

    Chandler Davis and William Morton Kahan. The rotation of eigenvectors by a perturbation, III . SIAM Journal on Numerical Analysis, 7 0 (1): 0 1--46, 1970

    work page 1970

  14. [14]

    The optimal hard threshold for singular values is 4/ 3

    David Donoho and Matan Gavish. The optimal hard threshold for singular values is 4/ 3 . IEEE Transactions on Information Theory, 60 0 (8): 0 5040--5053, 2014

    work page 2014

  15. [15]

    Donoho, Matan Gavish, and Iain M

    David L. Donoho, Matan Gavish, and Iain M. Johnstone. Optimal shrinkage of eigenvalues in the spiked covariance model. Annals of Statistics, 46 0 (4): 0 1742--1778, 2018

    work page 2018

  16. [16]

    Operator norm consistent estimation of large-dimensional sparse covariance matrices

    Noureddine El Karoui. Operator norm consistent estimation of large-dimensional sparse covariance matrices. Annals of Statistics, 36 0 (6): 0 2717--2756, 2008

    work page 2008

  17. [17]

    Eigen analysis of estimated covariance in high dimensions

    Jianqing Fan, Fang Jiao, and Jianfeng Yao. Eigen analysis of estimated covariance in high dimensions. Journal of Machine Learning Research, 19 0 (83): 0 1--50, 2018

    work page 2018

  18. [18]

    Relative perturbation bounds with applications to empirical covariance operators

    Moritz Jirak and Martin Wahl. Relative perturbation bounds with applications to empirical covariance operators. Advances in Mathematics, 412: 0 108808, 2023

    work page 2023

  19. [19]

    Quantitative limit theorems and bootstrap approximations for empirical spectral projectors

    Moritz Jirak and Martin Wahl. Quantitative limit theorems and bootstrap approximations for empirical spectral projectors. Probability Theory and Related Fields, 190: 0 119--177, 2024

    work page 2024

  20. [20]

    Johnstone

    Iain M. Johnstone. On the distribution of the largest eigenvalue in principal components analysis. Annals of Statistics, 29 0 (2): 0 295--327, 2001

    work page 2001

  21. [21]

    Normal approximation and concentration of spectral projectors of sample covariance

    Vladimir Koltchinskii and Karim Lounici. Normal approximation and concentration of spectral projectors of sample covariance. Annals of Statistics, 45 0 (1): 0 121--157, 2017

    work page 2017

  22. [22]

    Principal components as a measure of systemic risk

    Mark Kritzman, Yuanzhen Li, Sebasti \'a n Page, and Roberto Rigobon. Principal components as a measure of systemic risk. Journal of Portfolio Management, 37 0 (4): 0 112--126, 2011

    work page 2011

  23. [23]

    Eigenvectors of some large sample covariance matrix ensembles

    Olivier Ledoit and Sandrine P\'ech\'e. Eigenvectors of some large sample covariance matrix ensembles. Probability Theory and Related Fields, 151: 0 233--264, 2011

    work page 2011

  24. [24]

    A well-conditioned estimator for large-dimensional covariance matrices

    Olivier Ledoit and Michael Wolf. A well-conditioned estimator for large-dimensional covariance matrices. Journal of Multivariate Analysis, 88 0 (2): 0 365--411, 2004

    work page 2004

  25. [25]

    Analytical nonlinear shrinkage of large-dimensional covariance matrices

    Olivier Ledoit and Michael Wolf. Analytical nonlinear shrinkage of large-dimensional covariance matrices. Annals of Statistics, 48 0 (5): 0 3043--3065, 2020

    work page 2020

  26. [26]

    The performance of optimal estimation of large covariance matrices

    Olivier Ledoit and Michael Wolf. The performance of optimal estimation of large covariance matrices. Bernoulli, 28 0 (4): 0 2942--2970, 2022

    work page 2022

  27. [27]

    Eigenvector overlaps in large sample covariance matrices and nonlinear shrinkage estimators

    Zeqin Lin and Guangming Pan. Eigenvector overlaps in large sample covariance matrices and nonlinear shrinkage estimators. Annals of Statistics, 54 0 (2): 0 882--909, 2026

    work page 2026

  28. [28]

    V. A. Marchenko and L. A. Pastur. Distribution of eigenvalues for some sets of random matrices. Matematicheskii Sbornik, 72 0 (114): 0 507--536, 1967

    work page 1967

  29. [29]

    Muirhead

    Robb J. Muirhead. Aspects of Multivariate Statistical Theory. Wiley, New York, 1982

    work page 1982

  30. [30]

    Bootstrap confidence sets for spectral projectors of sample covariance

    Alexey Naumov, Vladimir Spokoiny, and Vladimir Ulyanov. Bootstrap confidence sets for spectral projectors of sample covariance. Probability Theory and Related Fields, 174: 0 1091--1132, 2019

    work page 2019

  31. [31]

    A canonical definition of shape

    Davy Paindaveine. A canonical definition of shape. Statistics & Probability Letters, 78 0 (14): 0 2240--2247, 2008

    work page 2008

  32. [32]

    Rank tests for PCA under weak identifiability

    Davy Paindaveine, Laura Peralvo Maroto , and Thomas Verdebout. Rank tests for PCA under weak identifiability. Annals of Statistics, 54 0 (2): 0 543--569, 2026

    work page 2026

  33. [33]

    On estimation of the noise variance in high dimensional probabilistic principal component analysis

    Damien Passemier, Zhaoyuan Li, and Jianfeng Yao. On estimation of the noise variance in high dimensional probabilistic principal component analysis. Journal of the Royal Statistical Society, Series B, 79 0 (1): 0 51--67, 2017

    work page 2017

  34. [34]

    Asymptotics of sample eigenstructure for a large dimensional spiked covariance model

    Debashis Paul. Asymptotics of sample eigenstructure for a large dimensional spiked covariance model. Statistica Sinica, 17 0 (4): 0 1617--1642, 2007

    work page 2007

  35. [35]

    Inference on the proportion of variance explained in principal component analysis

    Ronan Perry, Snigdha Panigrahi, Jacob Bien, and Daniela Witten. Inference on the proportion of variance explained in principal component analysis. Journal of the American Statistical Association, 121 0 (553): 0 667--677, 2025

    work page 2025

  36. [36]

    James R. Schott. Testing the equality of the smallest latent roots of a correlation matrix. Biometrika, 75 0 (4): 0 794--796, 1988

    work page 1988

  37. [37]

    Alexander Shapiro and Michael W. Browne. Analysis of covariance structures under elliptical distributions. Journal of the American Statistical Association, 82 0 (400): 0 1092--1097, 1987

    work page 1987

  38. [38]

    G. W. Stewart and Ji-guang Sun. Matrix Perturbation Theory. Academic Press, Boston, 1990

    work page 1990

  39. [39]

    David E. Tyler. Asymptotic inference for eigenvectors. Annals of Statistics, 9 0 (4): 0 725--736, 1981

    work page 1981

  40. [40]

    David E. Tyler. Robustness and efficiency properties of scatter matrices. Biometrika, 70 0 (2): 0 411--420, 1983

    work page 1983

  41. [41]

    High-Dimensional Probability: An Introduction with Applications in Data Science

    Roman Vershynin. High-Dimensional Probability: An Introduction with Applications in Data Science. Cambridge University Press, Cambridge, 2018

    work page 2018

  42. [42]

    Samworth

    Yi Yu, Tengyao Wang, and Richard J. Samworth. A useful variant of the Davis--Kahan theorem for statisticians. Biometrika, 102 0 (2): 0 315--323, 2015

    work page 2015