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For Gaussian-copula data, the normal-scores covariance inherits the latent correlation's spectral law, making nonlinear shrinkage asymptotically optimal and giving a closed-form spike-detection threshold.

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2026-08-01 11:35 UTC pith:NCZWQI6O

load-bearing objection Clean idea, plausible results, but the proof of the key perturbation lemma has a false independence claim and the eigenvector transfer is under-supported; worth a serious referee. the 4 major comments →

arxiv 2607.19825 v1 pith:NCZWQI6O submitted 2026-07-22 stat.ME math.STstat.TH

Mens: Nonlinear shrinkage estimation in nonparanormal models for financial applications

Hamid Karamikabir (1) , Mohammad Arashi (2) ((1) Department of Statistics , Faculty of Intelligent Systems Engineering , Data Science , Persian Gulf University , Bushehr , Iran , (2) Department of Statistics
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Faculty of Mathematical Sciences Ferdowsi University of Mashhad Mashhad Iran)
This is my paper
classification stat.ME math.STstat.TH MSC 62H1260B2062H2062P05
keywords nonparanormalGaussian copulanonlinear shrinkagenormal-scores rank covariancegeneralized Marchenko-Pastur lawspiked covariancehigh-dimensional covariance estimationportfolio optimization
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 paper tries to establish that in nonparanormal (Gaussian-copula) models, where each observed coordinate is an unknown strictly increasing transform of a latent Gaussian vector, the rank-based normal-scores covariance matrix behaves spectrally exactly like a Gaussian sample covariance with the latent correlation as its population covariance. Its empirical eigenvalue distribution therefore converges almost surely to the generalized Marchenko-Pastur law of the latent correlation, with the same limit for every choice of marginal transformations. Applying oracle nonlinear shrinkage to this matrix is then asymptotically optimal among all rotation-equivariant estimators under Frobenius loss, and spiked latent correlations separate from the bulk at a closed-form threshold. A sympathetic reader would care because this gives heavy-tailed, asymmetric financial data a covariance estimator that is simultaneously robust to unknown margins and theoretically efficient; the paper's S&P 500 backtest reports lower realized volatility, better conditioning, and lower turnover than linear shrinkage.

Core claim

Under the nonparanormal model, the oracle normal scores z*_ij = Phi^{-1}(F_j(x_ij)) exactly recover the latent Gaussian values y_ij, because F_j(x_ij) = Phi(y_ij). Hence the oracle covariance S*_n = n^{-1} sum_i y_i y_i^T is a genuine Gaussian sample covariance with population latent correlation Sigma, and the rank-based normal-scores covariance bS_n differs from it by o_P(1) in operator norm. The paper proves (Theorem 3.4) that the empirical spectral distribution of bS_n converges almost surely to the generalized Marchenko-Pastur law of Sigma, independent of the marginal transformations; that the oracle nonlinear shrinkage function minimizing asymptotic Frobenius risk among rotation-equivar

What carries the argument

The load-bearing object is the normal-scores covariance bS_n, obtained by replacing each column's values with Phi^{-1}(rank/(n+1)). Under the Gaussian-copula assumption, this transform inverts the unknown marginals exactly, so bS_n is an o_P(1) operator-norm perturbation of a Gaussian sample covariance with population latent correlation Sigma. The generalized Marchenko-Pastur equation—the Stieltjes-transform fixed-point system for the limiting spectrum—then supplies the oracle shrinkage function d*(lambda) = lambda / |1 - gamma - gamma lambda m_bar(lambda)|^2, and the boundary Stieltjes transform m_bar gives the spike-detection threshold and outlier map.

Load-bearing premise

The central proofs assume the normal-scores covariance matrix is o_P(1) close in operator norm to the oracle Gaussian sample covariance (Lemma 3.1); the proof treats the rows of the Hajek projection as i.i.d., but each entry depends on the whole sample through the empirical CDF, so the rows are not independent, and the eigenvector-level phase transition is transferred using only operator-norm closeness, which does not by itself control individual eigenvector quadratic forms.

What would settle it

Simulate p=200, n=400 under an identity-bulk Gaussian copula with a single subcritical spike theta = sqrt(gamma) - epsilon; if the largest eigenvalue of the normal-scores covariance systematically separates from the bulk edge, or if the squared cosine between the leading sample eigenvector and the spike direction stays above about 0.05 as n doubles, then the asserted phase transition and eigenvector alignment are false.

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

If this is right

  • Within the nonparanormal class, the spectrum of the rank-based covariance is invariant to the unknown marginal transformations, so the estimator is stable under skewness and heavy tails without losing the spectral theory on which shrinkage depends.
  • No rotation-equivariant estimator built from the normal-scores covariance can beat the oracle nonlinear shrinkage in asymptotic Frobenius risk, and the feasible MENS estimator attains that oracle risk.
  • Spiked latent correlations are detectable from the largest eigenvalue of the rank covariance precisely when the spike exceeds sqrt(gamma) for an identity bulk; below that threshold no eigenvalue-based test is consistent, and the threshold does not depend on the marginals.
  • The estimator is operator-norm consistent on the bulk, and the paper establishes a parametric n^{-1/2} minimax lower bound for bounded-spectrum correlation classes.
  • In the reported S&P 500 minimum-variance backtest, MENS reduces annualized out-of-sample portfolio volatility from 11.0% to 9.4%, improves the median condition number of the correlation estimate by more than a factor of three, and cuts turnover by 41% relative to linear shrinkage.

Where Pith is reading between the lines

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

  • If the latent distribution is not Gaussian, the exact recovery identity z*_ij = y_ij fails, and the limiting spectrum of the normal-scores covariance would be driven by a nonlinearly transformed latent variable rather than by Sigma; a diagnostic for Gaussian-copula structure could indicate when MENS is trustworthy.
  • The paper's Conjecture 6.3 suggests the sin-transformed Kendall matrix may share the same limiting spectrum even though its operator-norm distance to bS_n does not vanish; if true, MENS could be implemented directly on the Kendall matrix, avoiding normal-score inversion and possibly improving finite-sample behavior.
  • For p/n near 1, the threshold sqrt(gamma) approaches 1, so only very strong latent correlations can be separated from the bulk; a factor-augmented variant that shrinks only the bulk while fixing the spike via the outlier map would be a natural next estimator.
  • A direct test outside the model would be to simulate a t-copula with non-Gaussian dependence and measure the bias of the MENS spectrum relative to the generalized Marchenko-Pastur law; this would quantify the cost of the Gaussian-copula assumption.

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

4 major / 5 minor

Summary. The paper proposes MENS, a nonlinear shrinkage estimator of the latent correlation matrix in nonparanormal (Gaussian-copula) models. The estimator applies the Ledoit–Wolf oracle nonlinear shrinkage function to the normal-scores covariance matrix bSn, whose entries are rank-based transforms of the observed coordinates. The main theoretical claims are: (i) the empirical spectral distribution of bSn converges almost surely to the generalized Marchenko–Pastur law of the latent correlation Sigma, with the same limit for every choice of monotone marginals (Theorem 3.4); (ii) MENS is asymptotically optimal among rotation-equivariant estimators under Frobenius loss and achieves an operator-norm minimax lower bound of order n^{-1/2} (Theorem 3.9); and (iii) a Baik–Ben Arous–Péché phase transition with closed-form threshold sqrt(gamma), outlier map, and eigenvector alignment for spiked latent correlations (Theorem 3.10). The proof strategy is to compare bSn with the oracle Gaussian sample covariance S*_n via an operator-norm perturbation lemma (Lemma 3.1), and then transfer classical Gaussian random-matrix results. Simulations and an S&P 500 backtest are provided. The central theoretical results, however, rest on two unproved and, as written, incorrect proof steps: the row-independence claim in Lemma 3.1 and the operator-norm-to-eigenvector transfer in Lemmas 3.8 and 3.10.

Significance. If the central theorems were established, this would be a valuable contribution: it would extend the Ledoit–Wolf nonlinear shrinkage theory to a semiparametric Gaussian-copula setting, unifying the robustness of rank-based estimation with the efficiency of nonlinear shrinkage. The exact-recovery identity for oracle normal scores is elegant, and the paper clearly identifies the precise role of the Gaussian-copula assumption. The simulation study is well designed and the publicly available reference implementation is a strength. However, the main theorems are not proven by the arguments given. The proof of Lemma 3.1 relies on a false independence assumption, and the eigenvector-level results are transferred from S*_n to bSn using only operator-norm closeness, which is insufficient in the bulk. The paper itself concedes (Remark 3.5, Conjecture 6.1) that no local law is proved, and Conjecture 6.3 shows that a related operator-norm equivalence fails. Thus the three headline results—spectral limit, oracle optimality, and phase transition—are not established in the present manuscript.

major comments (4)
  1. [§3.1, Lemma 3.1, Eq. (16)] The proof claims that the rows of the Hájek projection matrix L are i.i.d. and applies Vershynin Thm 4.6.1 to √n L. This is false as written: each entry L_ij = g'_n(F_j(x_ij))(Phat_nj(x_ij)-F_j(x_ij)) depends on the full empirical CDF of column j, so L_ij and L_i'j are dependent for i ≠ i'. The cited row-independent matrix deviation inequality cannot be applied. Consequently the bound ||L||_op = O_P(√p/√n) and hence the o_P(1) operator-norm closeness of bSn to S*_n, on which Theorems 3.4, 3.9, and 3.10 all depend, is not established.
  2. [§3.3, Lemma 3.8 and Theorem 3.9] Lemma 3.8 attempts to prove convergence of the eigenvector quadratic form Rgamma_j^Rtop Σ Rgamma_j for bSn. The proof invokes the Ledoit–Péché deterministic equivalent for S*_n and a contour around Rlambda_j, but Rlambda_j is an eigenvalue of bSn while the resolvent is that of S*_n. The argument does not transfer eigenvector information from S*_n to bSn. Operator-norm closeness alone does not control individual eigenvector projections when bulk spacings are O(1/p); a local law would be required, and Remark 3.5 and Conjecture 6.1 concede that none is proved. Thus the projection limit (30) and the oracle optimality claim in Theorem 3.9(i) are unsupported.
  3. [§3.4, Theorem 3.10] The phase-transition proof argues with S*_n and asserts that the o_P(1) operator-norm perturbation from Lemma 3.1 transfers both eigenvalue outlier locations and eigenvector alignments. For eigenvalue outliers, Weyl's inequality gives closeness of leading eigenvalues, so the outlier location φ(θ) may be salvageable. However, the subcritical 'sticking to the edge' claim and the alignment limit (38) require control of individual eigenvectors of bSn. The Woodbury identity is applied to the finite-rank perturbation of S*_n; the perturbation bSn - S*_n is full-rank and not controlled at eigenvector level. No local law for bSn is supplied. Hence Theorem 3.10(ii) and the identity-bulk cosine formula (39) are not proven.
  4. [§2.3 and proof of Theorem 3.9(i)] The consistency of the feasible analytic shrinkage estimators (11)–(13) is imported from Ledoit and Wolf (2020, Thm 3.1). That theorem is stated for the sample covariance matrix of i.i.d. observations, whereas bSn has dependent rows (the normal scores use ranks computed from the full sample). Even if the eigenvalue-limiting law is the same, the conditions of the cited theorem need to be verified for bSn. As written, the assertion supi |Rd_i - d*(Rlambda_i)| = o_P(1) is an unexamined transfer and is load-bearing for the claim that the feasible MENS attains the oracle risk.
minor comments (5)
  1. [Throughout] Typos: 'becasue' in Section 3 heading; 'seperation' in Lemma 3.3; 'Mar cenko' spacing in several places. The typesetting of Rbar m and Rbreve m in Proposition 2.5 could be clarified.
  2. [Assumption A4] The parenthetical 'This is the only role A1 plays in the perturbation bound' is confusing because A1 is the nonparanormal model itself, not a regularity condition. Consider rewording.
  3. [Section 4.4 and Table 2] The GMV risk-ratio result shows linear shrinkage beating MENS on that metric; the authors explain this, but it weakens the practical motivation for the simulated setting. This is a presentation point, not a flaw.
  4. [Section 5.1] The admission that the real-data application lies outside the theoretical setting is welcome. However, the claim that the volatility ranking reverses at p/n ≳ 1 is only 'unreported experiments'; naming the experiments or deferring them would strengthen reproducibility.
  5. [Conjecture 6.3] This conjecture itself undermines the reliance on operator-norm equivalence for rank inputs. It would be helpful to state explicitly which of the paper's results would survive if only entrywise consistency, rather than operator-norm closeness, held.

Circularity Check

0 steps flagged

No significant circularity: the central results derive from an exact normal-score identity plus standard external random-matrix theorems; flagged limitations are proof gaps, not circularity.

full rationale

The derivation chain is not circular. The central reduction (Lemma 3.1) bounds ||bSn - S*_n||_op by an o_P(1) rank-perturbation argument; this is an independent mathematical claim, not an assumption of the target results. Theorem 3.4 then uses the exact-recovery identity z*_ij = y_ij (Remark 2.4, Eq. 7) to identify S*_n with a Gaussian sample covariance with population Sigma, and invokes the external Silverstein-Bai theorem. The oracle optimality of Theorem 3.9 follows from the Ledoit-Peche deterministic equivalent (31) and the Frobenius-loss expansion (34); no parameter is fitted to data and then renamed a prediction. The BBP phase transition of Theorem 3.10 is derived by Woodbury/deterministic-equivalent arguments and explicitly matches the classical Gaussian result, which is the correct consequence of exact score recovery. There are no self-citations by the present authors; all load-bearing citations are to standard external works (Silverstein-Bai, Ledoit-Wolf, Ledoit-Peche, Benaych-Georges-Nadakuditi, Bai-Silverstein). The paper itself flags unresolved technical limitations: Remark 3.5 disclaims a local law, Conjecture 6.1 asks for a quantitative rate, and Conjecture 6.3 admits that the operator-norm equivalence between bSn and the Kendall matrix is not proved. These are correctness/rigor concerns, not circularity: they do not assume the theorems being proved. The suspicious i.i.d.-row claim inside Lemma 3.1 is a potential proof gap, but it is not a circular step; the lemma is not equivalent to the target result by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 8 axioms · 0 invented entities

The central mathematical content rests on standard Gaussian random matrix theory plus the nonparanormal model assumption. No new physical or probabilistic entities are postulated. The only hand-chosen tuning quantity is the bandwidth scale inherited from Ledoit-Wolf (2020). The main unstated fragility is the validity of Lemma 3.1's operator-norm bound and the transfer of eigenvector-level results to bSn.

free parameters (1)
  • Epanechnikov bandwidth scale h_j = λhat_j n^{-1/3} = n^{-1/3}
    Hand-chosen bandwidth inherited from Ledoit-Wolf (2020) for analytic shrinkage density estimation. It is not fitted to the data, but the consistency of the feasible MENS estimator in Theorem 3.9(i) relies on the properties of this bandwidth.
axioms (8)
  • domain assumption A1: Data are exactly nonparanormal: x_j = f_j(y_j), y ~ N(0,Σ), with unknown strictly increasing f_j.
    This is the model assumption that makes the oracle normal scores exactly equal the latent Gaussian vector, so S*_n is a Gaussian sample covariance. It is introduced in Section 2.1 and is the load-bearing model throughout.
  • domain assumption A2: The ESD H_p of the latent correlation Σ converges weakly to a compactly supported measure H on (0,∞), with spectrum bounded away from 0 and ∞.
    Standard random-matrix condition needed for the Marchenko-Pastur law, the boundedness of the oracle shrinkage function, and the minimax class.
  • standard math A3: p/n → γ ∈ (0,∞) as min(n,p) → ∞.
    The proportional-growth regime assumed for all asymmetric random matrix limits.
  • domain assumption A5: Spike directions {v_k} are delocalized: max_k ||v_k||_∞ → 0.
    Required for the BBP outlier and eigenvector alignment analysis in Theorem 3.10.
  • standard math Silverstein-Bai Marchenko-Pastur theorem for Gaussian sample covariance matrices.
    Used directly in Theorem 3.4 Stage B to obtain the spectral limit for S*_n.
  • standard math Ledoit-Péché deterministic equivalent for weighted resolvents and Ledoit-Wolf oracle optimality for nonlinear shrinkage.
    Used in Lemma 3.8 and Theorem 3.9 to derive the oracle shrinkage function and the optimality claim.
  • standard math Baik-Ben Arous-Péché / Benaych-Georges-Nadakuditi spiked covariance results.
    Used in Theorem 3.10 to identify the outlier location and eigenvector alignment formulas for the oracle Gaussian matrix.
  • standard math Dvoretzky-Kiefer-Wolfowitz inequality and matrix deviation inequality for sub-Gaussian matrices.
    Used in Lemma 3.1 to bound the rank-perturbation error. The application of the matrix deviation inequality is itself questionable because the rows of L are not independent.

pith-pipeline@v1.3.0-alltime-deepseek · 23189 in / 15700 out tokens · 157109 ms · 2026-08-01T11:35:00.224437+00:00 · methodology

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read the original abstract

We develop a theory of nonlinear shrinkage covariance estimation for nonparanormal (Gaussian-copula) models, in which each observed coordinate is an unknown strictly increasing transformation of a latent Gaussian vector. This model accommodates arbitrary marginal skewness and heavy marginal tails while retaining a Gaussian dependence structure, and it is the natural semiparametric setting for heavy-tailed, asymmetric financial returns. Our estimator, marginal-free nonlinear shrinkage (MENS), applies an oracle nonlinear shrinkage function to the eigenvalues of the normal-scores rank-covariance matrix. We give the almost-sure convergence of the empirical spectral distribution of the normal-scores covariance to the generalized Marchenko-Pastur law of Sigma, and asymptotic optimality of MENS among rotation-equivariant estimators under Frobenius loss. We establish a Baik-Ben Arous-Peche phase transition for spiked latent correlations. The MENS attains the robustness of rank-based estimation and the efficiency of nonlinear shrinkage at once within this class. We corroborate the theory with a simulation study that isolates the marginal-invariance property and the spiked transition. In an out-of-sample minimum-variance backtest on S&P 500 stocks, MENS delivers a better-conditioned covariance estimate, lower realized portfolio volatility, and lower turnover than linear shrinkage, illustrating its practical value for high-dimensional allocation and decision-making.

Figures

Figures reproduced from arXiv: 2607.19825 by (2) Department of Statistics, Bushehr, Data Science, Faculty of Intelligent Systems Engineering, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Hamid Karamikabir (1), Iran, Iran), Mashhad, Mohammad Arashi (2) ((1) Department of Statistics, Persian Gulf University.

Figure 1
Figure 1. Figure 1: Marginal invariance. A single latent Gaussian sample [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Empirical spectral density of Sbn (identity latent correlation, γ = 0.5) for the mixed and logistic marginal banks at two sizes, with the standard Marcenko–Pastur density overlaid (red). The fit is the same across banks and ˇ tightens with n. The uncorrected Kendall matrix, lacking a spectral correction, is comparatively poorly conditioned and is the worst in operator norm. To illustrate the dependence on … view at source ↗
Figure 3
Figure 3. Figure 3: Replication-level loss distributions at (p, n) = (150, 300) over B = 60 replications: (a) operator-norm loss, (b) Frobenius loss. MENS (green) has the lowest median and a tight spread in both panels. the uncorrected Kendall matrix, being ill-conditioned, produces unusable portfolios (median 6.9 with a long tail). Also, note that the linear shrinkage toward the identity (LW) achieves a slightly lower risk r… view at source ↗
Figure 4
Figure 4. Figure 4: Spiked transition on nonparanormal data, [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Global minimum-variance risk ratio (log scale, oracle [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: S&P 500 minimum-variance backtest, p = 200, one-year rolling windows. (a) Cumulative out-of-sample wealth of the GMV portfolio (gross of transaction costs); the MENS curve is visibly smoother, reflecting its lower realized volatility (9.4% vs 11.0% annualized). (b) Rolling condition number of the estimated correlation matrix (log scale); MENS (green) is consistently better conditioned than linear shrinkage… view at source ↗

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Reference graph

Works this paper leans on

35 extracted references

  1. [1]

    and SILVERSTEIN, J

    BAI, Z. and SILVERSTEIN, J. W. (2010).Spectral Analysis of Large Dimensional Random Matrices, 2nd ed. Springer, New York

    2010

  2. [2]

    BAI, Z. D. and SILVERSTEIN, J. W. (1998). No eigenvalues outside the support of the limiting spectral distribution of large-dimensional sample covariance matrices.The Annals of Probability26(1) 316–345

    1998

  3. [3]

    BAI, Z. D. and YIN, Y. Q. (1993). Limit of the smallest eigenvalue of a large dimensional sample covariance matrix. The Annals of Probability21(3) 1275–1294

    1993

  4. [4]

    BAIK, J., BENAROUS, G., and PÉCHÉ, S. (2005). Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices.The Annals of Probability33(5) 1643–1697

    2005

  5. [5]

    S., LODHIA, A., and RIGOLLET, P

    BANDEIRA, A. S., LODHIA, A., and RIGOLLET, P. (2017). Mar ˇcenko–Pastur law for Kendall’s tau.Electronic Communications in Probability22(32) 1–7

    2017

  6. [6]

    BAO, Z. (2019). Tracy–Widom limit for Kendall’s tau.The Annals of Statistics47(6) 3504–3532

    2019

  7. [7]

    and NADAKUDITI, R

    BENAYCH-GEORGES, F. and NADAKUDITI, R. R. (2012). The singular values and vectors of low rank perturbations of large rectangular random matrices.Journal of Multivariate Analysis111120–135

    2012

  8. [8]

    CONT, R. (2001). Empirical properties of asset returns: Stylized facts and statistical issues.Quantitative Finance1(2) 223–236

    2001

  9. [9]

    and MCKAY, M

    COUILLET, R. and MCKAY, M. (2014). Large dimensional analysis and optimization of robust shrinkage covariance matrix estimators.Journal of Multivariate Analysis13199–120

    2014

  10. [10]

    EARLE, C. J. and HAMILTON, R. S. (1970). A fixed point theorem for holomorphic mappings.Proceedings of Symposia in Pure Mathematics1661–65

    1970

  11. [11]

    ELKAROUI, N. (2009). Concentration of measure and spectra of random matrices: Applications to correlation matrices, elliptical distributions and beyond.The Annals of Applied Probability19(6) 2362–2405

    2009

  12. [12]

    J., and STRAUMANN, D

    EMBRECHTS, P., MCNEIL, A. J., and STRAUMANN, D. (2002). Correlation and dependence in risk management: Properties and pitfalls. In DEMPSTER, M. A. H., editor,Risk Management: Value at Risk and Beyond, pages 176–223. Cambridge University Press

    2002

  13. [13]

    FAN, J., LIU, H., and WANG, W. (2018). Large covariance estimation through elliptical factor models.The Annals of Statistics46(4) 1383–1414

    2018

  14. [14]

    FRIEDMAN, J., HASTIE, T., and TIBSHIRANI, R. (2008). Sparse inverse covariance estimation with the graphical lasso. Biostatistics9(3) 432–441. HÁJEK, J. and ŠIDÁK, Z. (1967).Theory of Rank Tests. Academic Press, New York

    2008

  15. [15]

    and LIU, H

    HAN, F. and LIU, H. (2018). ECA: High-dimensional elliptical component analysis in non-Gaussian distributions. Journal of the American Statistical Association113(521) 252–268

    2018

  16. [16]

    JOHNSTONE, I. M. (2001). On the distribution of the largest eigenvalue in principal components analysis.The Annals of Statistics29(2) 295–327

    2001

  17. [17]

    KRUSKAL, W. H. (1958). Ordinal measures of association.Journal of the American Statistical Association53(284)

    1958

  18. [18]

    and PÉCHÉ, S

    LEDOIT, O. and PÉCHÉ, S. (2011). Eigenvectors of some large sample covariance matrix ensembles.Probability Theory and Related Fields151(1–2) 233–264

    2011

  19. [19]

    and WOLF, M

    LEDOIT, O. and WOLF, M. (2004). A well-conditioned estimator for large-dimensional covariance matrices.Journal of Multivariate Analysis88(2) 365–411

    2004

  20. [20]

    and WOLF, M

    LEDOIT, O. and WOLF, M. (2012). Nonlinear shrinkage estimation of large-dimensional covariance matrices.The Annals of Statistics40(2) 1024–1060

    2012

  21. [21]

    and WOLF, M

    LEDOIT, O. and WOLF, M. (2015). Spectrum estimation: A unified framework for covariance matrix estimation and PCA in large dimensions.Journal of Multivariate Analysis139360–384

    2015

  22. [22]

    and WOLF, M

    LEDOIT, O. and WOLF, M. (2020). Analytical nonlinear shrinkage of large-dimensional covariance matrices.The Annals of Statistics48(5) 3043–3065

    2020

  23. [23]

    and WOLF, M

    LEDOIT, O. and WOLF, M. (2022). Quadratic shrinkage for large covariance matrices.Bernoulli28(3) 1519–1547

    2022

  24. [24]

    LI, Z., WANG, C., and WANG, Q. (2023). On eigenvalues of a high-dimensional Kendall’s rank correlation matrix with dependence.Science China Mathematics66(11) 2615–2640

    2023

  25. [25]

    LI, Z., WANG, Q., and LI, R. (2021). Central limit theorem for linear spectral statistics of large dimensional Kendall’s rank correlation matrices and its applications.The Annals of Statistics49(3) 1569–1593. 20 MENS: Nonlinear Shrinkage EstimationA PREPRINT

    2021

  26. [26]

    LIU, H., HAN, F., YUAN, M., LAFFERTY, J., and WASSERMAN, L. (2012). High-dimensional semiparametric Gaussian copula graphical models.The Annals of Statistics40(4) 2293–2326

    2012

  27. [27]

    LIU, H., LAFFERTY, J., and WASSERMAN, L. (2009). The nonparanormal: Semiparametric estimation of high dimensional undirected graphs.Journal of Machine Learning Research102295–2328

    2009

  28. [28]

    MARKOWITZ, H. (1952). Portfolio selection.The Journal of Finance7(1) 77–91. MAR ˇCENKO, V. A. and PASTUR, L. A. (1967). Distribution of eigenvalues for some sets of random matrices. Mathematics of the USSR-Sbornik1(4) 457–483

    1952

  29. [29]

    MASSART, P. (1990). The tight constant in the Dvoretzky–Kiefer–Wolfowitz inequality.The Annals of Probability 18(3) 1269–1283

    1990

  30. [30]

    PAUL, D. (2007). Asymptotics of sample eigenstructure for a large dimensional spiked covariance model.Statistica Sinica17(4) 1617–1642

    2007

  31. [31]

    SILVERSTEIN, J. W. and BAI, Z. D. (1995). On the empirical distribution of eigenvalues of a class of large dimensional random matrices.Journal of Multivariate Analysis54(2) 175–192

    1995

  32. [32]

    TSYBAKOV, A. B. (2009).Introduction to Nonparametric Estimation. Springer, New York

    2009

  33. [33]

    (2018).High-Dimensional Probability: An Introduction with Applications in Data Science

    VERSHYNIN, R. (2018).High-Dimensional Probability: An Introduction with Applications in Data Science. Cambridge University Press

    2018

  34. [34]

    and ZHAO, Y

    WEGKAMP, M. and ZHAO, Y. (2016). Adaptive estimation of the copula correlation matrix for semiparametric elliptical copulas.Bernoulli22(2) 1184–1226

    2016

  35. [35]

    and WANG, C

    WU, Z. and WANG, C. (2022). Limiting spectral distribution of large dimensional Spearman’s rank correlation matrices. Journal of Multivariate Analysis191105011. 21

    2022