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REVIEW 3 major objections 5 minor 79 references

Application of Random Matrix Theory in High-Dimensional Statistics

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper proves a Berry–Esseen-type bound for the log-determinant of a Wishart matrix and uses it to build tests for covariance matrices in high dimensions.

desk verdict A useful RMT review whose one original theorem has a real scaling error in the proof; as stated, the theorem is false outside the p/√n → 0 regime, but the applications already live in that regime and the fix is straightforward. read the letter →

arxiv 2412.06848 v1 pith:5UPYXDC7 submitted 2024-12-08 stat.ME math.STstat.TH

Swapnaneel Bhattacharyya , Srijan Chattopadhyay , Sevantee Basu This is my paper · ORCID
classification stat.MEmath.STstat.TH MSC 60B2062H1062H15
keywords RandomMatrixTheoryEmpiricalSpectralDistributionLimitingWishartBerry-Esseenboundcovariancetestinghigh-dimensionalstatistics
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This is a review of random matrix theory as a toolbox for high-dimensional statistics, and it carries one original result of its own. Theorem 1 states that for a Wishart matrix $X_n \sim W_p(\Sigma,n)$, the centered and scaled sum of log-eigenvalue ratios — equivalently the log-determinant of $X_n$ relative to $\Sigma$ — converges to a standard normal with a Berry–Esseen error of order $p/\sqrt{n}$. A sympathetic reader should care because that rate turns a limit theorem into a usable approximation: the paper shows how it yields asymptotically size-$\alpha$ tests for $H_0:\Sigma=\Sigma_0$ in one-sample, regression, and two-sample problems, and how it gives an error bound for estimating functions of the population spectrum by sample eigenvalues. The rest of the article surveys the Marcenko–Pastur law, extreme-eigenvalue Tracy–Widom limits, spiked covariance phase transitions, and $F$-matrix results, and applies them to PCA, signal detection, and changepoint detection.

What carries the argument

The argument runs on the classical determinant factorization of a Wishart matrix: $|X_n| = |\Sigma| U_1\cdots U_p$, where $U_i \sim \chi^2_{n-p+i}$ are independent (the $i$-th factor carries the degrees of freedom $n-p+i$). This reduces the log-determinant statistic to a sum of independent log-chi-square terms, and the proof assembles a Berry–Esseen bound for each term (Lemma 3, proved from a delta-method bound of Pinelis and Molzon and the classical Berry–Esseen theorem) with a subadditivity lemma for sup-norm CDF distance (Lemma 1). The machinery delivers the $O(p/\sqrt{n})$ rate by adding $p$ terms each of order $1/\sqrt{n-p+i}$.

What would settle it

Simulate the statistic in Theorem 1 for a fixed ratio $p/n$ (say $p/n=1/2$) with large $n$, and compare the empirical CDF to $\Phi$. If the error does not shrink at rate $p/\sqrt{n}$, or the normalization is visibly wrong, the theorem's stated rate fails. More directly, inspect the proof's step where $\sqrt{n/2}$ is factored out of each log term: the omitted factor $\sqrt{n/(n-p+i)}$ differs from $1$ by a factor of about $\sqrt{2}$ at $i=1$ when $p/n=1/2$, so Lemma 3 cannot be applied as written; a simulation at that ratio would show whether the CLT itself survives the mismatch.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 1: if $X_n \sim W_p(\Sigma,n)$ with $n \ge p$, then $\sup_x |P( \sqrt{n/(2p)} (\sum_i \log(\lambda_i^{(n)}/\lambda_i) - \sum_i \log(n-p+i)) \le x) - \Phi(x)| = O(p/\sqrt{n})$. The statement is a CLT with explicit rate for the log-determinant of the Wishart matrix: the random part is exactly $(1/2)\sum_i \log(U_i/(n-p+i))$ with $U_i$ independent $\chi^2$ variables, so the theorem gives the fluctuation scale $n/(2p)$ and the centering $\sum_i \log(n-p+i)$. Section 4.1 converts the theorem into concrete testing procedures: an asymptotically size-$\alpha$ test of covariance equality in equation (4.4), its version for the sum-of-squares error matrix in high-dimensional regression in equation (4.6), and a two-sample test with an explicit approximate power function in equation (4.8). The paper claims these tests work when $p/\sqrt{n} \to 0$.

Load-bearing premise

The proof treats each term $\sqrt{n/2}\,\log(U_i/(n-p+i))$ as if the chi-square variable had $n$ degrees of freedom rather than $n-p+i$, an identification that is accurate only when $p$ is small relative to $n$.

Editorial extensions

If this is right

  • The statistic in Theorem 1 gives an asymptotically size-$\alpha$ test of $H_0:\Sigma=\Sigma_0$ with critical values from the standard normal (equation 4.4).
  • The same theorem supplies a test for the error covariance matrix in high-dimensional linear regression using the SSE matrix, which is Wishart under the null (equation 4.6).
  • For two independent samples, the theorem yields an asymptotic test of $\Sigma_1=\Sigma_2$ and an explicit formula for its power function in terms of the population eigenvalues (equation 4.8).
  • The $O(p/\sqrt{n})$ rate provides a bound on the error when functions of the population spectrum are approximated by sample eigenvalues, which the paper notes is useful for sample-size determination.

Reading between the lines

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

  • The theorem's regime $p/\sqrt{n} \to 0$ is much more restrictive than the usual $p/n \to \gamma$ regime of random matrix theory: $p$ must be asymptotically smaller than $\sqrt{n}$, so the tests are not directly usable in the standard proportional-growth setting.
  • If the rate $O(p/\sqrt{n})$ is sharp, the normal approximation should degrade once $p$ is comparable to $\sqrt{n}$; a natural extension would be a non-normal limiting law in which the scaling factor $\sqrt{n/(n-p+i)}$ is kept inside each log term.
  • The same machinery could be applied to other smooth functions of Wishart eigenvalues besides the logarithm, such as $\log\det(I+S)$ in signal-processing capacity computations, giving explicit convergence rates for those statistics.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper is a review of random matrix theory (RMT) in high-dimensional statistics, covering the spectral properties of sample covariance matrices and F-type matrices and surveying applications in covariance inference, PCA, signal processing, and changepoint detection. Its claimed original contribution is Theorem 1, a Berry-Esseen-type bound for the log-determinant of a Wishart matrix, with applications to one-sample, regression, and two-sample covariance tests in Section 4.1. The proof of Theorem 1 is given in the appendix.

Significance. If Theorem 1 held as stated, it would provide an explicit O(p/sqrt(n)) rate of convergence for a log-determinant CLT and would support new hypothesis tests for covariance matrices. The paper also compiles several classical RMT results and presents them in a statistical context, which is useful for a review. However, the theorem is false as stated without a p/sqrt(n) -> 0 restriction, and the proof omits a nontrivial scaling factor. The test statistics derived from the theorem in Section 4.1 contain additional centering errors, so the claimed applications are not valid in their current form. The survey component is reasonable, but the original contribution needs substantial correction.

major comments (3)
  1. [Section 6, Appendix (proof of Theorem 1)] Lemma 3 is applied to the term sqrt(n/2) log(U_i/(n-p+i)), but Lemma 3 controls sqrt((n-p+i)/2) log(U_i/(n-p+i)). The omitted factor sqrt(n/(n-p+i)) is not close to 1 uniformly in i unless p/n -> 0; its deviation is O((p-i)/n), which sums to O(p^2/n). For p/n -> gamma > 0, the variance of the normalized statistic is approximately (n/(2p)) * sum_i Var(log(U_i/(n-p+i))) ~ (1/gamma) log(1/(1-gamma)), which is not 1, so the standard normal limit in Theorem 1 is false for such sequences. The theorem must be restated with an explicit p/sqrt(n) -> 0 restriction (or another correct regime), and the proof must bound the distributional distance between the scaled summands and N(0,1) rather than invoking Lemma 3 directly. In addition, the displayed inequality after the reduction to Y_1+...+Y_p does not follow from Lemma 1 as written: the right-hand side still contains the same p-dimensional probability, so Lemma 1 must be applied successively to the individual summands, with each individual term stated explicitly.
  2. [Section 4.1, Eq. (4.4)] The test in Eq. (4.4) is not a valid application of Theorem 1. With S defined as (1/(n-1)) * sum (X_i - bar X)(X_i - bar X)^T and hat lambda_i its eigenvalues, the Wishart matrix is (n-1)S, not S. Theorem 1 applied to (n-1)S gives the statistic sqrt((n-1)/(2p)) [sum_i log(hat lambda_i/lambda_i) + p log(n-1) - sum_i log((n-1)-p+i)]. The expression in Eq. (4.4) omits the term p log(n-1) and uses log(n-p+i) instead of log((n-1)-p+i), leaving a deterministic offset of order sqrt((n-1)/(2p)) [p log(n-1) + log(n/(n-p))], which diverges as p, n -> infinity. Hence Eq. (4.4) does not define an asymptotically size-alpha test under the stated definitions.
  3. [Section 4.1, Eq. (4.8)] The two-sample test in Eq. (4.8) has an analogous centering error. Under H0, (m-1)S_X ~ W_p(Sigma, m-1) and (n-1)S_Y ~ W_p(Sigma, n-1), so the log-ratio of the Wishart determinants contains an additional p log((m-1)/(n-1)) term, and the centering should be sum_i log(((m-1)-p+i)/((n-1)-p+i)). Eq. (4.8) instead uses sum_i log((m-p+i)/(n-p+i)) and omits the p log((m-1)/(n-1)) term. For n/m -> c with c != 1, the omitted term times the normalization sqrt(m/(2p)(1+1/c)) is of order sqrt(mp) log c and does not vanish, so the proposed test is not asymptotically size alpha. The normalization also does not match the variance of the centered log-ratio.
minor comments (5)
  1. [Section 4.1, Eq. (4.6)] The index k in Eq. (4.6) is undefined, and the dimension index is inconsistent: the notation writes hat lambda_1, ..., hat lambda_p even though the SSE matrix is m x m. The correct centering should be sum_{i=1}^m log((n-r)-m+i) rather than sum log(n-r-k+i).
  2. [Section 4.4, Proposition 1] Proposition 1 refers to T as defined in (2.5), but the definition of T appears in Eq. (4.23); the cross-reference is incorrect.
  3. [Section 4.4, Theorem 12] The Lindeberg-type conditions in Theorem 12 are labeled (3.7) and (3.8), but they appear in Section 4.4; the equation numbering should be corrected.
  4. [Section 3.1.1] The sentence 'For gamma = 0 ... the maximum and minimum eigenvalues converge to 1' is imprecise: it should refer to the support of the limiting spectral distribution, or specify that this concerns the sample eigenvalues under a further scaling.
  5. [Throughout] There are several typographical and formatting errors, including 'eignevalues' in Section 3.1, 'Tthe generalized factor model' in the reference Forni et al., and inconsistent use of p versus m in parts of Section 4.1. A careful proofreading pass is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the new Theorem 1 is derived from standard external ingredients (Wishart determinant factorization, Berry-Esseen, Pinelis-Molzon), and the Section 4.1 tests are consequences of Theorem 1 rather than inputs to it.

full rationale

The paper's original contribution is Theorem 1, a Berry-Esseen-type bound for the log-determinant of a Wishart matrix. Its proof (Section 6) starts from the exact Wishart determinant decomposition |X_n| = |Σ| U_1 ... U_p with independent chi-square factors U_j ~ χ²_{n-p+j}, then bounds the Kolmogorov distance between the normalized log-determinant and the standard normal CDF using Lemma 1 (a triangle inequality for CDF distances), Lemma 2 (a bound derived from Pinelis and Molzon (2016)), and Lemma 3 (a Berry-Esseen bound for log chi-square variables derived from Lemma 2 and the classical Berry-Esseen theorem). These are standard, independent external results; the proof does not assume Theorem 1 or any of the Section 4.1 test statistics as an input. The applications in Section 4.1 define tests (4.4), (4.6), and (4.8) by applying Theorem 1 to Wishart matrices; these are consequences, not premises, of the theorem. There are no fitted parameters, no uniqueness claims imported from the authors' prior work, and no self-citations are load-bearing (the reference list contains no self-citations by the authors). The proof does contain a likely mathematical gap: Lemma 3 is stated for sqrt(m/2) log(Z_m/m), while in the theorem the term sqrt(n/2) log(U_i/(n-p+i)) is used, so the factor sqrt(n/(n-p+i)) is omitted; this could make the stated O(p/sqrt(n)) bound invalid in regimes where p is comparable to n. That is a correctness or validity concern, not circular reasoning. The derivation chain does not reduce to its own inputs, so the circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The new theorem relies on standard probabilistic results and does not introduce free parameters, novel entities, or ad hoc assumptions. The ledger is therefore minimal, but the proof has a scaling gap that is not an axiom but a derivation error.

assumptions (4)
  • standard math Bartlett decomposition: for W_p(Σ, n), |W| = |Σ| ∏_{j=1}^p χ²_{n-j+1} with independent chi-square variables.
    Used in the proof of Theorem 1 in the appendix, at the step after Lemma 3 where they write '|Xn| = |Σ|U1···Up'.
  • standard math Berry-Esseen theorem for sums of i.i.d. variables.
    Used in Lemma 3 to bound the distance between the log-chi-square distribution and the normal.
  • standard math Delta method bound from Pinelis and Molzon (2016), Theorem 2.10.
    Used in Lemma 2 to obtain the O(1/√m) rate for the transformed sample mean.
  • standard math Convolution inequality for Kolmogorov distance (Lemma 1).
    Used in the proof of Theorem 1 to decompose the distance for a sum of independent terms.

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Cite this review

Pith. "Pith review of Application of Random Matrix Theory in High-Dimensional Statistics." pith.science (2026). https://pith.science/paper/5UPYXDC7

@misc{pith2026241206848,
  author       = {Pith},
  title        = {Pith review of: Application of Random Matrix Theory in High-Dimensional Statistics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5UPYXDC7}},
  note         = {Machine review of arXiv:2412.06848}
}
read the original abstract

This review article provides an overview of random matrix theory (RMT) with a focus on its growing impact on the formulation and inference of statistical models and methodologies. Emphasizing applications within high-dimensional statistics, we explore key theoretical results from RMT and their role in addressing challenges associated with high-dimensional data. The discussion highlights how advances in RMT have significantly influenced the development of statistical methods, particularly in areas such as covariance matrix inference, principal component analysis (PCA), signal processing, and changepoint detection, demonstrating the close interplay between theory and practice in modern high-dimensional statistical inference.

Figures

Figures reproduced from arXiv: 2412.06848 by the authors.

Figure 1
Figure 1. Marcenko–Pastur density functions for γ = 0.1, 0.25, 0.5, 1 where b±(γ) = (1 ± √ γ) 2 . For x outside this interval, fγ(x) = 0. If γ ∈ (1,∞), then Fγ is a mixture of a point mass at 0 and the p.d.f. f1/γ(x), with weights 1 − 1/γ and 1/γ, respectively. It is to be noted that the above result is distribution-free, in the sense the limiting distribution only depends on the limiting ratio of data dimension and sample si… view at source ↗
Figure 2
Figure 2. Density function for the Tracy-Widom Distribution [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
Figure 3
Figure 3. (Paul (2007)) An illustration of the phase transition of eigenvalues in a spiked covariance model: here, p = 50, n = 200 and eigenvalues of the covariance matrix are ℓ1 = 2.5, ℓ2 = 1.5, ℓj = 1 for j = 3, . . . , p. So, ℓ1 > 1 + p p/n and ℓ2 = 1 + p p/n. Blue dots correspond to the population eigenvalues. Black circles correspond to the sample eigenvalues (based on i.i.d. Gaussian samples) for 50 replicates. Solid re… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Number of Estimated Signals for different noise distributions: (in clockwise order) [PITH_FULL_IMAGE:figures/full_fig_p037_4.png]

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Works this paper leans on

79 extracted references · 60 canonical work pages

  1. [1]

    write newline

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  2. [2]

    @esa (Ref

    \@ifclassloaded agu2001 natbib The agu2001 class already includes natbib coding, so you should not add it explicitly Type <Return> for now, but then later remove the command natbib from the document \@ifclassloaded aguplus natbib The aguplus class already includes natbib coding, so you should not add it explicitly Type <Return> for now, but then later rem...

  3. [3]

    @stdbsttrue NAT@ctr \@lbibitem[ NAT@ctr ] \@lbibitem[#1]#2 \@extra@b@citeb \@ifundefined br@#2\@extra@b@citeb \@namedef br@#2 \@nameuse br@#2\@extra@b@citeb \@ifundefined b@#2\@extra@b@citeb @num @parse #2 [ @natanchorstart #2\@extra@b@citeb \@biblabel @num @natanchorend] @ifcmd#1(@)(@)\@nil #2 @lbibitem\@undefined @lbibitem\@lbibitem \@lbibitem[#1]#2 @lb...

  4. [4]

    @open @close @open @close and [1] URL: #1 \@ifundefined chapter * \@mkboth \@ifundefined NAT@sectionbib * \@mkboth * \@mkboth\@gobbletwo \@ifclassloaded amsart * \@ifclassloaded amsbook * \@ifundefined bib@heading @heading NAT@ctr thebibliography [1] @ \@biblabel NAT@ctr \@bibsetup #1 NAT@ctr 0 @openbib .11em \@plus.33em \@minus.07em 4000 4000 `\.=1000 \@...

    work page

  5. [5]

    and Cook, D

    Aminikhanghahi, S. and Cook, D. J. (2017). A survey of methods for time series change point detection. Knowledge and information systems , 51(2):339--367

    work page 2017

  6. [6]

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

    1963

  7. [7]

    and Buzun, N

    Avanesov, V. and Buzun, N. (2018). Change-point detection in high-dimensional covariance structure

    work page 2018

  8. [8]

    and Silverstein, J

    Bai, Z. and Silverstein, J. W. (2007). On the signal-to-interference ratio of cdma systems in wireless communications

    work page 2007

Show all 79 references
  1. [9]

    and Silverstein, J

    Bai, Z. and Silverstein, J. W. (2010). Spectral analysis of large dimensional random matrices , volume 20. Springer

    2010

  2. [10]

    and Zhou, W

    Bai, Z. and Zhou, W. (2008). Large sample covariance matrices without independence structures in columns. Statistica Sinica , pages 425--442

    2008

  3. [11]

    Bai, Z. D. and Yin, Y. Q. (1988). Convergence to the semicircle law. The Annals of Probability , 16(2):863--875

    1988

  4. [12]

    and Yin, Y.-Q

    Bai, Z.-D. and Yin, Y.-Q. (2008). Limit of the smallest eigenvalue of a large dimensional sample covariance matrix. In Advances In Statistics , pages 108--127. World Scientific

    2008

  5. [13]

    D., Yin, Y

    Bai, Z. D., Yin, Y. Q., and Krishnaiah, P. R. (1988). On the limiting empirical distribution function of the eigenvalues of a multivariate f matrix. Theory of Probability & Its Applications , 32(3):490--500

    1988

  6. [14]

    and Silverstein, J

    Baik, J. and Silverstein, J. W. (2006). Eigenvalues of large sample covariance matrices of spiked population models. Journal of multivariate analysis , 97(6):1382--1408

    2006

  7. [15]

    Banerjee, T., Firouzi, H., and Hero, A. O. (2015). Non-parametric quickest change detection for large scale random matrices. In 2015 IEEE International Symposium on Information Theory (ISIT) , pages 146--150. IEEE

    2015

  8. [16]

    Bao, Z. (2012). Strong convergence of esd for the generalized sample covariance matrices when p/n→ 0. Statistics & Probability Letters , 82(5):894--901

    2012

  9. [17]

    Billingsley, P. (2013). Convergence of probability measures . John Wiley & Sons

    2013

  10. [18]

    and Debbah, M

    Couillet, R. and Debbah, M. (2011). Random matrix methods for wireless communications . Cambridge University Press

    2011

  11. [19]

    and Liao, Z

    Couillet, R. and Liao, Z. (2022). Random matrix methods for machine learning . Cambridge University Press

    2022

  12. [20]

    Deb, S., Pourahmadi, M., and Wu, W. B. (2017). An asymptotic theory for spectral analysis of random fields

    2017

  13. [21]

    Finn, J. D. (1974). A general model for multivariate analysis. Holt, Rinehart & Winston

    1974

  14. [22]

    Forni, M., Hallin, M., Lippi, M., and Reichlin, L. (2000). Tthe generalized factor model: identification and estimation. U\

    2000

  15. [23]

    Friesen, O., L \"o we, M., and Stolz, M. (2013). Gaussian fluctuations for sample covariance matrices with dependent data. Journal of Multivariate Analysis , 114:270--287

    2013

  16. [24]

    Fryzlewicz, P. (2014). Wild binary segmentation for multiple change-point detection

    2014

  17. [25]

    and Tikhomirov, A

    Gotze, F. and Tikhomirov, A. (2006). Limit theorems for spectra of positive random matrices under dependence. Journal of Mathematical Sciences , 133:1257--1276

    2006

  18. [26]

    Hachem, W., Loubaton, P., Mestre, X., Najim, J., and Vallet, P. (2013). A subspace estimator for fixed rank perturbations of large random matrices. Journal of Multivariate Analysis , 114:427--447

    2013

  19. [27]

    Han, X., Pan, G., and Zhang, B. (2016). The tracy--widom law for the largest eigenvalue of f type matrices

    2016

  20. [28]

    Hastie, T., Buja, A., and Tibshirani, R. (1995). Penalized discriminant analysis. The Annals of Statistics , 23(1):73--102

    1995

  21. [29]

    Haynes, W. (2013). Bonferroni correction. Encyclopedia of systems biology , pages 154--154

    2013

  22. [30]

    and Rajaratnam, B

    Hero, A. and Rajaratnam, B. (2012). Hub discovery in partial correlation graphs. IEEE Transactions on Information Theory , 58(9):6064--6078

    2012

  23. [31]

    and Stolz, M

    Hofmann-Credner, K. and Stolz, M. (2008). Wigner theorems for random matrices with dependent entries: ensembles associated to symmetric spaces and sample covariance matrices

    2008

  24. [32]

    and Rattray, M

    Hoyle, D. and Rattray, M. (2003). Limiting form of the sample covariance eigenspectrum in pca and kernel pca. Advances in Neural Information Processing Systems , 16

    2003

  25. [33]

    and Pan, G

    Hui, J. and Pan, G. (2010). Limiting spectral distribution for large sample covariance matrices with m-dependent elements. Communications in Statistics—Theory and Methods , 39(6):935--941

    2010

  26. [34]

    and Lee, H.-N

    James, O. and Lee, H.-N. (2014). Concise probability distributions of eigenvalues of real-valued wishart matrices. arXiv preprint arXiv:1402.6757

    2014 arXiv

  27. [35]

    Johnson, D. E. and Graybill, F. A. (1972). An analysis of a two-way model with interaction and no replication. Journal of the American Statistical Association , 67(340):862--868

    1972

  28. [36]

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

    2001

  29. [37]

    Johnstone, I. M. (2006). High dimensional statistical inference and random matrices. arXiv preprint math/0611589

    2006 arXiv

  30. [38]

    Johnstone, I. M. and Lu, A. Y. (2009). Sparse principal components analysis. arXiv preprint arXiv:0901.4392

    2009 arXiv

  31. [39]

    Johnstone, I. M. and Nadler, B. (2017). Roy’s largest root test under rank-one alternatives. Biometrika , 104(1):181--193

    2017

  32. [40]

    Karoui, N. E. (2003). On the largest eigenvalue of wishart matrices with identity covariance when n, p and p/n tend to infinity. arXiv preprint math/0309355

    2003 arXiv

  33. [41]

    and Nadler, B

    Kritchman, S. and Nadler, B. (2009). Non-parametric detection of the number of signals: Hypothesis testing and random matrix theory. IEEE Transactions on Signal Processing , 57(10):3930--3941

    2009

  34. [42]

    Lawley, D. N. (1938). A generalization of fisher's z test. Biometrika , 30(1/2):180--187

    1938

  35. [43]

    Ma, Z. (2012). Accuracy of the tracy--widom limits for the extreme eigenvalues in white wishart matrices

    2012

  36. [44]

    Marchenko, V. A. and Pastur, L. A. (1967). Distribution of eigenvalues for some sets of random matrices. Matematicheskii Sbornik , 114(4):507--536

    1967

  37. [45]

    V., Kent, J

    Mardia, K. V., Kent, J. T., and Taylor, C. C. (2024). Multivariate analysis , volume 88. John Wiley & Sons

    2024

  38. [46]

    Muirhead, R. J. (2009). Aspects of multivariate statistical theory . John Wiley & Sons

    2009

  39. [47]

    Nadakuditi, R. R. and Silverstein, J. W. (2010). Fundamental limit of sample generalized eigenvalue based detection of signals in noise using relatively few signal-bearing and noise-only samples. IEEE Journal of selected topics in Signal Processing , 4(3):468--480

    2010

  40. [48]

    Nadler, B. (2008). Finite sample approximation results for principal component analysis: A matrix perturbation approach

    2008

  41. [49]

    S., Hao, N., and Zhang, H

    Niu, Y. S., Hao, N., and Zhang, H. (2016). Multiple change-point detection: a selective overview. Statistical Science , pages 611--623

    2016

  42. [50]

    and Rubin, H

    Olkin, I. and Rubin, H. (1964). Multivariate beta distributions and independence properties of the wishart distribution. The Annals of Mathematical Statistics , pages 261--269

    1964

  43. [51]

    B., Venkatraman, E

    Olshen, A. B., Venkatraman, E. S., Lucito, R., and Wigler, M. (2004). Circular binary segmentation for the analysis of array-based dna copy number data. Biostatistics , 5(4):557--572

    2004

  44. [52]

    Paul, D. (2007). Asymptotics of sample eigenstructure for a large dimensional spiked covariance model. Statistica Sinica , pages 1617--1642

    2007

  45. [53]

    and Aue, A

    Paul, D. and Aue, A. (2014). Random matrix theory in statistics: A review. Journal of Statistical Planning and Inference , 150:1--29

    2014

  46. [54]

    Pearson, K. (1901). Liii. on lines and planes of closest fit to systems of points in space. The London, Edinburgh, and Dublin philosophical magazine and journal of science , 2(11):559--572

    1901

  47. [55]

    Perlman, M. D. (1977). A note on the matrix-variate f distribution. Sankhy \=a : The Indian Journal of Statistics, Series A , pages 290--298

    1977

  48. [56]

    and Molzon, R

    Pinelis, I. and Molzon, R. (2016). Optimal-order bounds on the rate of convergence to normality in the multivariate delta method

    2016

  49. [57]

    Potthoff, R. F. and Roy, S. N. (1964). A generalized multivariate analysis of variance model useful especially for growth curve problems. Biometrika , 51(3-4):313--326

    1964

  50. [58]

    Roy, S. N. (1953). On a heuristic method of test construction and its use in multivariate analysis. The Annals of Mathematical Statistics , 24(2):220--238

    1953

  51. [59]

    and Killick, R

    Ryan, S. and Killick, R. (2023). Detecting changes in covariance via random matrix theory. Technometrics , 65(4):480--491

    2023

  52. [60]

    Scott, A. J. and Knott, M. (1974). A cluster analysis method for grouping means in the analysis of variance. Biometrics , pages 507--512

    1974

  53. [61]

    Silverstein, J. W. (1995). Strong convergence of the empirical distribution of eigenvalues of large dimensional random matrices. Journal of Multivariate Analysis , 55(2):331--339

    1995

  54. [62]

    Silverstein, J. W. and Combettes, P. L. (1992). Signal detection via spectral theory of large dimensional random matrices. IEEE Transactions on Signal Processing , 40(8):2100--2105

    1992

  55. [63]

    Soshnikov, A. (2002). A note on universality of the distribution of the largest eigenvalues in certain sample covariance matrices. Journal of Statistical Physics , 108:1033--1056

    2002

  56. [64]

    Telatar, E. (1999). Capacity of multi-antenna gaussian channels. European transactions on telecommunications , 10(6):585--595

    1999

  57. [65]

    Tipping, M. E. and Bishop, C. M. (1999). Probabilistic principal component analysis. Journal of the Royal Statistical Society Series B: Statistical Methodology , 61(3):611--622

    1999

  58. [66]

    Tracy, C. A. and Widom, H. (1996). On orthogonal and symplectic matrix ensembles. Communications in Mathematical Physics , 177:727--754

    1996

  59. [67]

    M., Verd \'u , S., et al

    Tulino, A. M., Verd \'u , S., et al. (2004). Random matrix theory and wireless communications. Foundations and Trends in Communications and Information Theory , 1(1):1--182

    2004

  60. [68]

    Wachter, K. W. (1980). The limiting empirical measure of multiple discriminant ratios. The Annals of Statistics , pages 937--957

    1980

  61. [69]

    Wang, D., Yu, Y., and Rinaldo, A. (2017). Optimal covariance change point localization in high dimension. arXiv preprint arXiv:1712.09912

    2017 arXiv

  62. [70]

    and Kailath, T

    Wax, M. and Kailath, T. (1985). Detection of signals by information theoretic criteria. IEEE Transactions on acoustics, speech, and signal processing , 33(2):387--392

    1985

  63. [71]

    Wei, M., Yang, G., and Yang, L. (2016). The limiting spectral distribution for large sample covariance matrices with unbounded m-dependent entries. Communications in Statistics-Theory and Methods , 45(22):6651--6662

    2016

  64. [72]

    Wilks, S. S. (1932). Certain generalizations in the analysis of variance. Biometrika , pages 471--494

    1932

  65. [73]

    Wishart, J. (1928). The generalised product moment distribution in samples from a normal multivariate population. Biometrika , pages 32--52

    1928

  66. [74]

    Yao, J. (2012). A note on a mar c enko--pastur type theorem for time series. Statistics & probability letters , 82(1):22--28

    2012

  67. [75]

    Yin, Y., Bai, Z., and Krishnaiah, P. (1983). Limiting behavior of the eigenvalues of a multivariate f matrix. Journal of multivariate analysis , 13(4):508--516

    1983

  68. [76]

    Yin, Y., Bai, Z., and Krishnaiah, P. (1984). On limit of the largest eigenvalue of the large dimensional sample covariance matrix

    1984

  69. [77]

    Yin, Y. Q. (1986). Limiting spectral distribution for a class of random matrices. Journal of multivariate analysis , 20(1):50--68

    1986

  70. [78]

    Yin, Y. Q. and Krishnaiah, P. R. (1983). A limit theorem for the eigenvalues of product of two random matrices. Journal of Multivariate Analysis , 13(4):489--507

    1983

  71. [79]

    and Krishnaiah, P

    Yin, Y.-Q. and Krishnaiah, P. R. (1987). Limit theorems for the eigenvalues of a product of large dimensional random matrices when the underlying distribution is isotropic. Theory of Probability & Its Applications , 31(2):342--346

    1987

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