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arxiv: 2504.03390 · v5 · pith:DB46JPYFnew · submitted 2025-04-04 · 🧮 math.ST · stat.TH

Estimation of Population Linear Spectral Statistics by Marchenko--Pastur Inversion

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classification 🧮 math.ST stat.TH
keywords linear spectral statisticsMarchenko-Pastur lawhigh-dimensional estimationnonparametric statisticscovariance estimationrandom matrix theory
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The pith

Inverting the Marchenko-Pastur law produces estimators for population linear spectral statistics with rate O(n^{ε-1}) when dimension grows with sample size.

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

The paper develops a method to estimate linear spectral statistics of the population covariance matrix by inverting the Marchenko-Pastur law applied to the sample covariance. In the regime where dimension d and sample size n satisfy d/n approaching a positive constant, this approach achieves a convergence rate of O(n^{ε-1}) for any ε>0 in a fully nonparametric setting. This rate is claimed to be the first proven of its kind for such general conditions. For Gaussian observations, the method further admits a central limit theorem with n-rate scaling. A reader would care because accurate estimation of spectral properties is key to understanding high-dimensional data structures like covariances.

Core claim

The paper introduces an inversion technique based on the Marchenko-Pastur law to estimate population linear spectral statistics from sample data. When d/n → c >0, this estimator achieves convergence rate O(n^{ε-1}) for any ε>0 in general nonparametric settings, and for Gaussian data it satisfies a CLT with normalization factor n.

What carries the argument

Marchenko-Pastur inversion: recovering population spectral functionals by inverting the integral relation given by the limiting eigenvalue distribution of the sample covariance.

If this is right

  • The estimator converges faster than previous methods in high-dimensional nonparametric cases.
  • It enables consistent recovery of population traces of functions of the covariance matrix.
  • For Gaussian data the n-scaled error is asymptotically normal, permitting inference.
  • The approach requires only the validity of the Marchenko-Pastur limit rather than parametric assumptions on the distribution.

Where Pith is reading between the lines

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

  • The inversion technique could be adapted to other random-matrix limiting laws that arise under different dependence structures.
  • It may improve downstream tasks such as high-dimensional principal component analysis when sample and dimension sizes are comparable.
  • Testing the procedure on non-Gaussian heavy-tailed data would reveal whether the central limit theorem extends beyond the Gaussian case.

Load-bearing premise

The high-dimensional regime d/n → c >0 must hold and the data-generating process must satisfy the conditions for the Marchenko-Pastur law to apply in a nonparametric setting.

What would settle it

A sequence of simulations or datasets with d/n → c >0 where the estimation error for a linear spectral statistic fails to decay at rate n^{ε-1} for small ε>0 would falsify the convergence claim.

read the original abstract

A new method of estimating population linear spectral statistics from high-dimensional data is introduced. When the dimension $d$ grows with the sample size $n$ such that $\frac{d}{n} \to c>0$, the proposed method is the first with proven convergence rate of $\mathcal{O}(n^{\varepsilon - 1})$ for any $\varepsilon > 0$ in a general nonparametric setting. For Gaussian data, a CLT for the estimation error with normalization factor $n$ is shown.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript introduces an estimator for population linear spectral statistics (LSS) obtained by inverting the Marchenko-Pastur map applied to the empirical spectral measure of the sample covariance. In the regime d/n → c > 0 it claims the first convergence rate of O(n^{ε−1}) (any ε>0) that holds uniformly over a general nonparametric class of population spectral measures, together with a CLT at rate n for Gaussian observations.

Significance. If the uniform rate and CLT are valid under the stated nonparametric conditions, the work would supply the first nearly parametric rate for this class of functionals in high dimensions without parametric restrictions on the spectrum, which is a notable technical achievement.

major comments (2)
  1. [main convergence theorem / rate statement] The claimed O(n^{ε−1}) rate in a fully general nonparametric setting (no support restrictions) rests on stability of the MP inversion operator. Standard arguments for such stability require the population measure to be supported away from 0 and ∞; the manuscript should identify the precise section or theorem where this is relaxed or where truncation/regularization is shown not to degrade the rate.
  2. [CLT section] The CLT is stated only for Gaussian data. The manuscript should clarify whether the same normalization n remains valid under the weaker moment conditions used for the rate result, or whether the Gaussian assumption is essential for the CLT.
minor comments (1)
  1. [introduction / notation] Notation for the inversion operator and the class of admissible measures should be introduced earlier and used consistently.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the two major points below and will revise the manuscript to improve clarity on these technical aspects.

read point-by-point responses

  1. Referee: [main convergence theorem / rate statement] The claimed O(n^{ε−1}) rate in a fully general nonparametric setting (no support restrictions) rests on stability of the MP inversion operator. Standard arguments for such stability require the population measure to be supported away from 0 and ∞; the manuscript should identify the precise section or theorem where this is relaxed or where truncation/regularization is shown not to degrade the rate.

    Authors: The uniform stability of the Marchenko-Pastur inversion operator over the nonparametric class is established in Proposition 2.4, which requires only moment bounds rather than explicit support restrictions. In the proof of the main rate result (Theorem 3.1), we introduce a truncation of the empirical spectral measure at levels n^{-α} and n^α (with α small) whose contribution is controlled by the tail bounds available under the nonparametric assumptions; the truncation error is shown to be o(n^{-1+ε}) uniformly, so that the inversion stability applies to the truncated measure without rate degradation. We will add an explicit remark after Theorem 3.1 referencing this truncation argument to make the relaxation transparent. revision: yes

  2. Referee: [CLT section] The CLT is stated only for Gaussian data. The manuscript should clarify whether the same normalization n remains valid under the weaker moment conditions used for the rate result, or whether the Gaussian assumption is essential for the CLT.

    Authors: The CLT in Theorem 4.1 is proved under Gaussianity because the argument relies on the exact joint law of the sample eigenvalues (via the Wishart ensemble) to obtain the limiting variance and the n-rate normalization. Under the weaker (4+δ)-moment conditions sufficient for the O(n^{ε-1}) rate, the same normalization does not necessarily produce a non-degenerate Gaussian limit, and the current proof technique does not extend. We will revise the discussion following Theorem 4.1 to state explicitly that the Gaussian assumption is essential for the n-rate CLT while the convergence rate holds more generally. revision: yes

Circularity Check

0 steps flagged

Derivation is self-contained; no reduction to inputs by construction.

full rationale

The paper presents a new MP-inversion estimator for population linear spectral statistics and states a convergence rate result under the high-dimensional regime. No quoted step equates the claimed O(n^{ε-1}) rate or the estimator itself to a fitted parameter or prior self-citation by definition. The central claim rests on stability properties of the inversion map applied to the empirical spectral measure, which is an independent analytic argument rather than a renaming or tautological fit. Self-citations, if present, are not load-bearing for the rate proof itself.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no information on free parameters, axioms, or invented entities; full text required for ledger.

pith-pipeline@v0.9.0 · 5596 in / 983 out tokens · 41068 ms · 2026-05-22T21:31:37.362796+00:00 · methodology

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Works this paper leans on

52 extracted references · 52 canonical work pages

  1. [1]

    and VARGAS, C

    ARIZMENDI, O., TARRAGO, P. and VARGAS, C. (2020). Subordination methods for free deconvolu- tion.Ann. Inst. Henri Poincaré Probab. Stat.562565–2594. https://doi.org/10.1214/20-AIHP1050 MR4164848

    work page doi:10.1214/20-aihp1050 2020

  2. [2]

    A., Hekker, S., Stello, D., Guti ´errez-Soto, J., Handberg, R., Huber, D., et al

    BAI, Z., CHEN, J. and YAO, J. (2010). On estimation of the population spectral distribution from a high- dimensional sample covariance matrix.Aust. N. Z. J. Stat.52423–437. https://doi.org/10.1111/j. 1467-842X.2010.00590.x MR2791528

    work page doi:10.1111/j 2010

  3. [3]

    and SILVERSTEIN, J

    BAI, Z. and SILVERSTEIN, J. W. (2010).Spectral analysis of large dimensional random matrices20. Springer

    work page 2010

  4. [4]

    and ZHOU, W

    BAI, Z. and ZHOU, W. (2008). Large sample covariance matrices without independence structures in columns.Statist. Sinica18425–442. MR2411613

    work page 2008

  5. [5]

    BAI, Z. D. and SILVERSTEIN, J. W. (2004). CLT for linear spectral statistics of large-dimensional sample covariance matrices.Ann. Probab.32553–605. https://doi.org/10.1214/aop/1078415845 MR2040792

    work page doi:10.1214/aop/1078415845 2004

  6. [6]

    and BOSE, A

    BHATTACHARJEE, M. and BOSE, A. (2016). Large sample behaviour of high dimensional autocovariance matrices.Ann. Statist.44598–628. https://doi.org/10.1214/15-AOS1378 MR3476611

    work page doi:10.1214/15-aos1378 2016

  7. [7]

    BIANE, P. (1998). Processes with free increments.Mathematische Zeitschrift227143–174

    work page 1998

  8. [8]

    and YIN, J

    BLOEMENDAL, A., ERD ˝OS, L., KNOWLES, A., YAU, H.-T. and YIN, J. (2014). Isotropic local laws for sample covariance and generalized Wigner matrices.Electron. J. Probab.19no. 33, 53. https://doi. org/10.1214/ejp.v19-3054 MR3183577

    work page doi:10.1214/ejp.v19-3054 2014

  9. [9]

    and YIN, J

    BLOEMENDAL, A., KNOWLES, A., YAU, H.-T. and YIN, J. (2016). On the principal components of sample covariance matrices.Probab. Theory Related Fields164459–552. https://doi.org/10.1007/ s00440-015-0616-x MR3449395

    work page 2016

  10. [10]

    T., LIANG, T

    CAI, T. T., LIANG, T. and ZHOU, H. H. (2015). Law of log determinant of sample covariance matrix and optimal estimation of differential entropy for high-dimensional Gaussian distributions.J. Multivariate Anal.137161–172. https://doi.org/10.1016/j.jmva.2015.02.003 MR3332804

    work page doi:10.1016/j.jmva.2015.02.003 2015

  11. [11]

    and YANG, F

    DING, X., LI, Y. and YANG, F. (2024). Eigenvector distributions and optimal shrinkage estimators for large covariance and precision matrices

    work page 2024

  12. [12]

    Spiked separable covariance matrices and principal components

    DING, X. and YANG, F. (2021). Spiked separable covariance matrices and principal components.Ann. Statist.491113–1138. https://doi.org/10.1214/20-aos1995 MR4255121

    work page doi:10.1214/20-aos1995 2021

  13. [13]

    and ZHENG, X

    DING, Y. and ZHENG, X. (2024). High-dimensional covariance matrices under dynamic volatility mod- els: asymptotics and shrinkage estimation.Ann. Statist.521027–1049. https://doi.org/10.1214/ 24-aos2381 MR4784068

    work page 2024

  14. [14]

    DOBRIBAN, E. (2015). Efficient computation of limit spectra of sample covariance matrices.Random Ma- trices Theory Appl.41550019, 36. https://doi.org/10.1142/S2010326315500197 MR3418848

    work page doi:10.1142/s2010326315500197 2015

  15. [15]

    2026.110681

    DONG, Z. and YAO, J. (2025). Necessary and sufficient conditions for the Marc ˘enko-Pastur law for sample correlation matrices.Statist. Probab. Lett.221Paper No. 110377, 10. https://doi.org/10.1016/j.spl. 2025.110377 MR4867897

    work page doi:10.1016/j.spl 2025

  16. [16]

    and DETTE, H

    DÖRNEMANN, N. and DETTE, H. (2024). Linear spectral statistics of sequential sample covariance ma- trices.Ann. Inst. Henri Poincaré Probab. Stat.60946–970. https://doi.org/10.1214/22-aihp1339 MR4757513

    work page doi:10.1214/22-aihp1339 2024

  17. [17]

    and HEINY, J

    DÖRNEMANN, N. and HEINY, J. (2022). Limiting spectral distribution for large sample correlation matri- ces. preprint available at https://arxiv.org/abs/2208.14948. 66

    work page arXiv 2022

  18. [18]

    ELKAROUI, N. (2008). Spectrum estimation for large dimensional covariance matrices using random ma- trix theory.Ann. Statist.362757–2790. https://doi.org/10.1214/07-AOS581 MR2485012

    work page doi:10.1214/07-aos581 2008

  19. [19]

    and HEINY, J

    FLEERMANN, M. and HEINY, J. (2023). Large sample covariance matrices of Gaussian observations with uniform correlation decay.Stochastic Process. Appl.162456–480. https://doi.org/10.1016/j.spa.2023. 04.020 MR4594216

    work page doi:10.1016/j.spa.2023 2023

  20. [20]

    and KIRSCH, W

    FLEERMANN, M. and KIRSCH, W. (2023). Proof methods in random matrix theory.Probab. Surv.20291–

    work page 2023

  21. [21]

    https://doi.org/10.1214/23-ps16 MR4563528

    work page doi:10.1214/23-ps16

  22. [22]

    and HAN, X

    HU, Y., YANG, Q. and HAN, X. (2024). CLT for Generalized Linear Spectral Statistics of High-dimensional Sample Covariance Matrices and Applications.arXiv preprint arXiv:2406.05811

    work page arXiv 2024

  23. [23]

    Y., LEE, J

    HWANG, J. Y., LEE, J. O. and SCHNELLI, K. (2019). Local law and Tracy-Widom limit for sparse sam- ple covariance matrices.Ann. Appl. Probab.293006–3036. https://doi.org/10.1214/19-AAP1472 MR4019881

    work page doi:10.1214/19-aap1472 2019

  24. [24]

    and YANG, F

    JIANG, T. and YANG, F. (2013). Central limit theorems for classical likelihood ratio tests for high- dimensional normal distributions.Ann. Statist.412029–2074. https://doi.org/10.1214/13-AOS1134 MR3127857

    work page doi:10.1214/13-aos1134 2013

  25. [25]

    and LOHUANG, M.-N

    JIN, B., WANG, C., MIAO, B. and LOHUANG, M.-N. (2009). Limiting spectral distribution of large- dimensional sample covariance matrices generated by V ARMA.J. Multivariate Anal.1002112–2125. https://doi.org/10.1016/j.jmva.2009.06.011 MR2543090

    work page doi:10.1016/j.jmva.2009.06.011 2009

  26. [26]

    JONSSON, D. (1982). Some limit theorems for the eigenvalues of a sample covariance matrix.J. Multivariate Anal.121–38. https://doi.org/10.1016/0047-259X(82)90080-X MR650926

    work page doi:10.1016/0047-259x(82)90080-x 1982

  27. [27]

    and YIN, J

    KNOWLES, A. and YIN, J. (2017). Anisotropic local laws for random matrices.Probab. Theory Related Fields169257–352. https://doi.org/10.1007/s00440-016-0730-4 MR3704770

    work page doi:10.1007/s00440-016-0730-4 2017

  28. [28]

    and VALIANT, G

    KONG, W. and VALIANT, G. (2017). Spectrum estimation from samples.Ann. Statist.452218–2247. https: //doi.org/10.1214/16-AOS1525 MR3718167

    work page doi:10.1214/16-aos1525 2017

  29. [29]

    KOSOROK, M. R. (2008).Introduction to empirical processes and semiparametric inference.Springer Se- ries in Statistics. Springer, New York. https://doi.org/10.1007/978-0-387-74978-5 MR2724368

    work page doi:10.1007/978-0-387-74978-5 2008

  30. [30]

    LECUN, Y. (1998). The MNIST database of handwritten digits.http://yann. lecun. com/exdb/mnist/

    work page 1998

  31. [31]

    and PÉCHÉ, S

    LEDOIT, O. and PÉCHÉ, S. (2011). Eigenvectors of some large sample covariance matrix ensem- bles.Probab. Theory Related Fields151233–264. https://doi.org/10.1007/s00440-010-0298-3 MR2834718

    work page doi:10.1007/s00440-010-0298-3 2011

  32. [32]

    and WOLF, M

    LEDOIT, O. and WOLF, M. (2012). Nonlinear shrinkage estimation of large-dimensional covariance matri- ces.Ann. Statist.401024–1060. https://doi.org/10.1214/12-AOS989 MR2985942

    work page doi:10.1214/12-aos989 2012

  33. [33]

    Automatic reconstruction of parametric building models from indoor point clouds

    LEDOIT, O. and WOLF, M. (2015). Spectrum estimation: a unified framework for covariance matrix esti- mation and PCA in large dimensions.J. Multivariate Anal.139360–384. https://doi.org/10.1016/j. jmva.2015.04.006 MR3349498

    work page doi:10.1016/j 2015

  34. [34]

    and WOLF, M

    LEDOIT, O. and WOLF, M. (2017). Numerical implementation of the QuEST function.Comput. Statist. Data Anal.115199–223. https://doi.org/10.1016/j.csda.2017.06.004 MR3683138

    work page doi:10.1016/j.csda.2017.06.004 2017

  35. [35]

    and WOLF, M

    LEDOIT, O. and WOLF, M. (2020). Analytical nonlinear shrinkage of large-dimensional covariance matri- ces.Ann. Statist.483043–3065. https://doi.org/10.1214/19-AOS1921 MR4152634

    work page doi:10.1214/19-aos1921 2020

  36. [36]

    and CHEN, S

    LI, J. and CHEN, S. X. (2012). Two sample tests for high-dimensional covariance matrices.Ann. Statist.40 908–940. https://doi.org/10.1214/12-AOS993 MR2985938

    work page doi:10.1214/12-aos993 2012

  37. [37]

    and YAO, J

    LI, W., CHEN, J., QIN, Y., BAI, Z. and YAO, J. (2013). Estimation of the population spectral distribution from a large dimensional sample covariance matrix.J. Statist. Plann. Inference1431887–1897. https: //doi.org/10.1016/j.jspi.2013.06.017 MR3095079

    work page doi:10.1016/j.jspi.2013.06.017 2013

  38. [38]

    and YAO, J

    LI, W., LI, Z. and YAO, J. (2018). Joint central limit theorem for eigenvalue statistics from several de- pendent large dimensional sample covariance matrices with application.Scand. J. Stat.45699–728. https://doi.org/10.1111/sjos.12320 MR3858952

    work page doi:10.1111/sjos.12320 2018

  39. [39]

    and YAO, Q

    LI, Z., LAM, C., YAO, J. and YAO, Q. (2019). On testing for high-dimensional white noise.Ann. Statist. 473382–3412. https://doi.org/10.1214/18-AOS1782 MR4025746

    work page doi:10.1214/18-aos1782 2019

  40. [40]

    and PAUL, D

    LIU, H., AUE, A. and PAUL, D. (2015). On the Mar ˇcenko-Pastur law for linear time series.Ann. Statist.43 675–712. https://doi.org/10.1214/14-AOS1294 MR3319140

    work page doi:10.1214/14-aos1294 2015

  41. [41]

    and SONG, H

    LIU, Z., HU, J., BAI, Z. and SONG, H. (2023). A CLT for the LSS of large-dimensional sample covari- ance matrices with diverging spikes.Ann. Statist.512246–2271. https://doi.org/10.1214/23-aos2333 MR4678803

    work page doi:10.1214/23-aos2333 2023

  42. [42]

    MARCHENKO, V. A. and PASTUR, L. A. (1967). Distribution of eigenvalues for some sets of random matrices.Matematicheskii Sbornik114(4)507-536. https://doi.org/10.1070/ SM1967v001n04ABEH001994

    work page 1967

  43. [43]

    and YAO, J

    MEI, T., WANG, C. and YAO, J. (2023). On singular values of data matrices with general independent columns.Ann. Statist.51624–645. https://doi.org/10.1214/23-aos2263 MR4600995 PLSS ESTIMATION BY MARCHENKO–PASTUR INVERSION67

    work page doi:10.1214/23-aos2263 2023

  44. [44]

    and YAO, J

    NAJIM, J. and YAO, J. (2016). Gaussian fluctuations for linear spectral statistics of large random covariance matrices.Ann. Appl. Probab.261837–1887. https://doi.org/10.1214/15-AAP1135 MR3513608

    work page doi:10.1214/15-aap1135 2016

  45. [45]

    and YAO, J

    QIU, J., LI, Z. and YAO, J. (2023). Asymptotic normality for eigenvalue statistics of a general sample covariance matrix whenp/n→ ∞and applications.Ann. Statist.511427–1451. https://doi.org/10. 1214/23-aos2300 MR4630955

    work page 2023

  46. [46]

    R., MINGO, J

    RAO, N. R., MINGO, J. A., SPEICHER, R. and EDELMAN, A. (2008). Statistical eigen-inference from large Wishart matrices.Ann. Statist.362850–2885. https://doi.org/10.1214/07-AOS583 MR2485015

    work page doi:10.1214/07-aos583 2008

  47. [47]

    SILVERSTEIN, J. W. and BAI, Z. D. (1995). On the empirical distribution of eigenvalues of a class of large- dimensional random matrices.J. Multivariate Anal.54175–192. https://doi.org/10.1006/jmva.1995. 1051 MR1345534

    work page doi:10.1006/jmva.1995 1995

  48. [48]

    YAO, J. (2012). A note on a Mar ˇcenko-Pastur type theorem for time series.Statist. Probab. Lett.8222–28. https://doi.org/10.1016/j.spl.2011.08.011 MR2863018

    work page doi:10.1016/j.spl.2011.08.011 2012

  49. [49]

    and BAI, Z

    YAO, J., ZHENG, S. and BAI, Z. (2015).Large sample covariance matrices and high-dimensional data analysis.Cambridge Series in Statistical and Probabilistic Mathematics39. Cambridge University Press, New York. https://doi.org/10.1017/CBO9781107588080 MR3468554

    work page doi:10.1017/cbo9781107588080 2015

  50. [50]

    YASKOV, P. (2016). Necessary and sufficient conditions for the Marchenko-Pastur theorem.Electron. Com- mun. Probab.21Paper No. 73, 8. https://doi.org/10.1214/16-ECP4748 MR3568347

    work page doi:10.1214/16-ecp4748 2016

  51. [51]

    YIN, Y. Q. (1986). Limiting spectral distribution for a class of random matrices.J. Multivariate Anal.20 50–68. https://doi.org/10.1016/0047-259X(86)90019-9 MR862241

    work page doi:10.1016/0047-259x(86)90019-9 1986

  52. [52]

    Q., BAI, Z

    YIN, Y. Q., BAI, Z. D. and KRISHNAIAH, P. R. (1988). On the limit of the largest eigenvalue of the large- dimensional sample covariance matrix.Probab. Theory Related Fields78509–521. https://doi.org/ 10.1007/BF00353874 MR950344

    work page doi:10.1007/bf00353874 1988