29916716
Hypergeometric functions of matrix arguments and linear statistics of multi-spiked Hermitian matrix models
This paper derives central limit theorems (CLTs) for general linear spectral statistics (LSS) of three important multi-spiked Hermitian random matrix ensembles. The first is the most common spiked scenario, proposed by Johnstone, which is a central Wishart ensemble with fixed-rank perturbation of the identity matrix, the second is a non-central Wishart ensemble with fixed-rank noncentrality parameter, and the third is a similarly defined non-central F ensemble. These CLT results generalize our recent work Passemier (2015) to account for multiple spikes, which is the most common scenario met in practice. The generalization is non-trivial, as it now requires dealing with hypergeometric functions of matrix arguments. To facilitate our analysis, for a broad class of such functions, we first generalize a recent result of Onatski (2014) to present new contour integral representations, which are particularly suitable for computing large-dimensional properties of spiked matrix ensembles. Armed with such representations, our CLT formulas are derived for each of the three spiked models of interest by employing the Coulomb fluid method from random matrix theory along with saddlepoint techniques. We find that for each matrix model, and for general LSS, the individual spikes contribute additively to yield a O ( 1 ) correction term to the asymptotic mean of the linear statistic, which we specify explicitly, whilst having no effect on the leading order terms of the mean or variance.
120743340
On the linear statistics of Hermitian random matrices
In this paper we continue with the study of linear statistics of random Hermitian matrix ensembles. We generalize the result of a previous paper on the probability density function of linear statistics to the finite N ensembles where the interval of support of the eigenvalue spectrum is a single interval. Combining the Hankel determinant formula for orthogonal polynomials which are associated with Hermitian matrix ensembles and the linear statistics theorem for finite N, we obtain the strong or oscillatory asymptotics for the polynomials orthogonal with respect to weight functions supported on the real axis.
15440253
CENTRAL LIMIT THEOREM FOR LINEAR EIGENVALUE STATISTICS OF RANDOM MATRICES WITH INDEPENDENT ENTRIES
We consider n x n real symmetric and Hermitian Wigner random matrices n ―1/2 W with independent (modulo symmetry condition) entries and the (null) sample covariance matrices n ―1 X*X with independent entries of m x n matrix X. Assuming first that the 4th cumulant (excess) κ 4 of entries of W and X is zero and that their 4th moments satisfy a Lindeberg type condition, we prove that linear statistics of eigenvalues of the above matrices satisfy the central limit theorem (CLT) as n → ∞, m → ∞, m/n → c ∈ [0, oo) with the same variance as for Gaussian matrices if the test functions of statistics are smooth enough (essentially of the class C 5 ). This is done by using a simple "interpolation trick" from the known results for the Gaussian matrices and the integration by parts, presented in the form of certain differentiation formulas. Then, by using a more elaborated version of the techniques, we prove the CLT in the case of nonzero excess of entries again for essentially ℂ 5 test function. Here the variance of statistics contains an additional term proportional to κ 4 . The proofs of all limit theorems follow essentially the same scheme.
6762379
A CLT for a band matrix model
A law of large numbers and a central limit theorem are derived for linear statistics of random symmetric matrices whose on-or-above diagonal entries are independent, but neither necessarily identically distributed, nor necessarily all of the same variance. The derivation is based on systematic combinatorial enumeration, study of generating functions, and concentration inequalities of the Poincaré type. Special cases treated, with an explicit evaluation of limiting variances, are generalized Wigner and Wishart matrices.
88513559
Asymptotic Linear Spectral Statistics for Spiked Hermitian Random Matrices
Using the Coulomb Fluid method, this paper derives central limit theorems (CLTs) for linear spectral statistics of three “spiked” Hermitian random matrix ensembles. These include Johnstone’s spiked model (i.e., central Wishart with spiked correlation), non-central Wishart with rank-one non-centrality, and a related class of non-central $$F$$F matrices. For a generic linear statistic, we derive simple and explicit CLT expressions as the matrix dimensions grow large. For all three ensembles under consideration, we find that the primary effect of the spike is to introduce an $$O(1)$$O(1) correction term to the asymptotic mean of the linear spectral statistic, which we characterize with simple formulas. The utility of our proposed framework is demonstrated through application to three different linear statistics problems: the classical likelihood ratio test for a population covariance, the capacity analysis of multi-antenna wireless communication systems with a line-of-sight transmission path, and a classical multiple sample significance testing problem.
118223160
Probability densities and distributions for spiked Wishart $\beta$-ensembles
A Wishart matrix is said to be spiked when the underlying covariance matrix has a single eigenvalue $b$ different from unity. As $b$ increases through $b=2$, a gap forms from the largest eigenvalue to the rest of the spectrum, and with $b-2$ of order $N^{-1/3}$ the scaled largest eigenvalues form a well defined parameter dependent state. Recent works by Bloemendal and Vir\'ag [BV], and Mo, have quantified this parameter dependent state for real Wishart matrices from different viewpoints, and the former authors have done similarly for the spiked Wishart $\beta$-ensemble. We use a recursive structure to give an alternative construction of the spiked Wishart $\beta$-ensemble, and we give the exact form of the joint eigenvalue PDF for the two matrices in the recurrence. In the case of real quaternion Wishart matrices ($\beta = 4$) the latter is recognised as having appeared in earlier studies on symmetrized last passage percolation, allowing the exact form of the scaled distribution of the largest eigenvalue to be given. This extends and simplifies earlier work of Wang, and is an alternative derivation to a result in [BV]. We also use the construction of the spiked Wishart $\beta$-ensemble from [BV] to give a simple derivation of the explicit form of the eigenvalue PDF.
55816065
The Largest Eigenvalue of Real Symmetric, Hermitian and Hermitian Self-dual Random Matrix Models with Rank One External Source, Part I
We consider the limiting location and limiting distribution of the largest eigenvalue in real symmetric (β=1), Hermitian (β=2), and Hermitian self-dual (β=4) random matrix models with rank 1 external source. They are analyzed in a uniform way by a contour integral representation of the joint probability density function of eigenvalues. Assuming the “one-band” condition and certain regularities of the potential function, we obtain the limiting location of the largest eigenvalue when the nonzero eigenvalue of the external source matrix is not the critical value, and further obtain the limiting distribution of the largest eigenvalue when the nonzero eigenvalue of the external source matrix is greater than the critical value. When the nonzero eigenvalue of the external source matrix is less than or equal to the critical value, the limiting distribution of the largest eigenvalue will be analyzed in a subsequent paper. In this paper we also give a definition of the external source model for all β>0.
11099902
Asymptotic power of sphericity tests for high-dimensional data
This paper studies the asymptotic power of tests of sphericity against perturbations in a single unknown direction as both the dimensionality of the data and the number of observations go to infinity. We establish the convergence, under the null hypothesis and the alternative, of the log ratio of the joint densities of the sample covariance eigenvalues to a Gaussian process indexed by the norm of the perturbation. When the perturbation norm is larger than the phase transition threshold studied in Baik et al. (2005), the limiting process is degenerate and discrimination between the null and the alternative is asymptotically certain. When the norm is below the threshold, the process is non-degenerate, so that the joint eigenvalue densities under the null and alternative hypotheses are mutually contiguous. Using the asymptotic theory of statistical experiments, we obtain asymptotic power envelopes and derive the asymptotic power for various sphericity tests in the contiguity region. In particular, we show that the asymptotic power of the Tracy-Widom-type tests is trivial, whereas that of the eigenvalue-based likelihood ratio test is strictly larger than the size, and close to the power envelope.
120291130
Financial Applications of Random Matrix Theory: a short review
We discuss the applications of Random Matrix Theory in the context of financial markets and econometric models, a topic about which a considerable number of papers have been devoted to in the last decade. This mini-review is intended to guide the reader through various theoretical results (the Marcenko-Pastur spectrum and its various generalisations, random SVD, free matrices, largest eigenvalue statistics, etc.) as well as some concrete applications to portfolio optimisation and out-of-sample risk estimation.
14795113
Corrections to LRT on large-dimensional covariance matrix by RMT
In this paper, we give an explanation to the failure of two likelihood ratio procedures for testing about covariance matrices from Gaussian populations when the dimension p is large compared to the sample size n. Next, using recent central limit theorems for linear spectral statistics of sample covariance matrices and of random F -matrices, we propose necessary corrections for these LR tests to cope with high-dimensional effects. The asymptotic distributions of these corrected tests under the null are given. Simulations demonstrate that the corrected LR tests yield a realized size close to nominal level for both moderate p (around 20) and high dimension, while the traditional LR tests with χ 2 approximation fails. Another contribution from the paper is that for testing the equality between two covariance matrices, the proposed correction applies equally for non-Gaussian populations yielding a valid pseudo-likelihood ratio test.
88518662
Testing linear hypotheses in high-dimensional regressions
For a multivariate linear model, Wilk's likelihood ratio test (LRT) constitutes one of the cornerstone tools. However, the computation of its quantiles under the null or the alternative hypothesis requires complex analytic approximations, and more importantly, these distributional approximations are feasible only for moderate dimension of the dependent variable, say p≤20. On the other hand, assuming that the data dimension p as well as the number q of regression variables are fixed while the sample size n grows, several asymptotic approximations are proposed in the literature for Wilk's Λ including the widely used chi-square approximation. In this paper, we consider necessary modifications to Wilk's test in a high-dimensional context, specifically assuming a high data dimension p and a large sample size n. Based on recent random matrix theory, the correction we propose to Wilk's test is asymptotically Gaussian under the null hypothesis and simulations demonstrate that the corrected LRT has very satisfactory size and power, surely in the large p and large n context, but also for moderately large data dimensions such as p=30 or p=50. As a byproduct, we give a reason explaining why the standard chi-square approximation fails for high-dimensional data. We also introduce a new procedure for the classical multiple sample significance test in multivariate analysis of variance which is valid for high-dimensional data.
11772431
Statistical Linkage Analysis of Substitutions in Patient-Derived Sequences of Genotype 1a Hepatitis C Virus Nonstructural Protein 3 Exposes Targets for Immunogen Design
ABSTRACT Chronic hepatitis C virus (HCV) infection is one of the leading causes of liver failure and liver cancer, affecting around 3% of the world's population. The extreme sequence variability of the virus resulting from error-prone replication has thwarted the discovery of a universal prophylactic vaccine. It is known that vigorous and multispecific cellular immune responses, involving both helper CD4+ and cytotoxic CD8+ T cells, are associated with the spontaneous clearance of acute HCV infection. Escape mutations in viral epitopes can, however, abrogate protective T-cell responses, leading to viral persistence and associated pathologies. Despite the propensity of the virus to mutate, there might still exist substitutions that incur a fitness cost. In this paper, we identify groups of coevolving residues within HCV nonstructural protein 3 (NS3) by analyzing diverse sequences of this protein using ideas from random matrix theory and associated methods. Our analyses indicate that one of these groups comprises a large percentage of residues for which HCV appears to resist multiple simultaneous substitutions. Targeting multiple residues in this group through vaccine-induced immune responses should either lead to viral recognition or elicit escape substitutions that compromise viral fitness. Our predictions are supported by published clinical data, which suggested that immune genotypes associated with spontaneous clearance of HCV preferentially recognized and targeted this vulnerable group of residues. Moreover, mapping the sites of this group onto the available protein structure provided insight into its functional significance. An epitope-based immunogen is proposed as an alternative to the NS3 epitopes in the peptide-based vaccine IC41. IMPORTANCE Despite much experimental work on HCV, a thorough statistical study of the HCV sequences for the purpose of immunogen design was missing in the literature. Such a study is vital to identify epistatic couplings among residues that can provide useful insights for designing a potent vaccine. In this work, ideas from random matrix theory were applied to characterize the statistics of substitutions within the diverse publicly available sequences of the genotype 1a HCV NS3 protein, leading to a group of sites for which HCV appears to resist simultaneous substitutions possibly due to deleterious effect on viral fitness. Our analysis leads to completely novel immunogen designs for HCV. In addition, the NS3 epitopes used in the recently proposed peptide-based vaccine IC41 were analyzed in the context of our framework. Our analysis predicts that alternative NS3 epitopes may be worth exploring as they might be more efficacious.
2038142
A note on the CLT of the LSS for sample covariance matrix from a spiked population model
In this note, we establish an asymptotic expansion for the centering parameter appearing in the central limit theorems for linear spectral statistic of large-dimensional sample covariance matrices when the population has a spiked covariance structure. As an application, we provide an asymptotic power function for the corrected likelihood ratio statistic for testing the presence of spike eigenvalues in the population covariance matrix. This result generalizes an existing formula from the literature where only one simple spike exists.
12793388
Detection of weak signals in high-dimensional complex-valued data
This paper considers the problem of detecting a few signals in high-dimensional complex-valued Gaussian data satisfying Johnstone's (2001) \textit{spiked covariance model}. We focus on the difficult case where signals are weak in the sense that the sizes of the corresponding covariance spikes are below the \textit{phase transition threshold} studied in Baik et al (2005). We derive a simple analytical expression for the maximal possible asymptotic probability of correct detection holding the asymptotic probability of false detection fixed. To accomplish this derivation, we establish what we believe to be a new formula for the \textit{% Harish-Chandra/Itzykson-Zuber (HCIZ) integral} $\int_{\mathcal{U}(p)}e^{\tr(AGBG^{-1})}dG $, where $A$ has a deficient rank $r<p$. The formula links the HCIZ integral over $\mathcal{U}(p) $ to an HCIZ integral over a potentially much smaller unitary group $\mathcal{U}(r) $. We show that the formula generalizes to the integrals over orthogonal and symplectic groups. In the most general form, it expresses the hypergeometric function $_{0}F_{0}^{(\alpha)}$of two $p\times p$ matrix arguments as a repeated contour integral of the hypergeometric function $_{0}F_{0}^{(\alpha)}$of two $r\times r$ matrix arguments.
15994360
Variance estimation and goodness-of-fit test in a high-dimensional strict factor model
In the classic setting where the dimension p is small compared to the sample size n, an asymptotic likelihood estimation theory is well-known for the factor model by letting n tending to in nity while keeping p fi xed. This theory is however no more valid for high-dimensional data where typically the dimension p is large compared to the sample size. In this paper, we develop new asymptotic results under the high-dimensional setting in a strict factor model with homoscedastic noise variance. For the maximum likelihood estimator of the noise variance, first we identify the reasons of a widely observed downward bias of the estimator. Second, a bias-corrected estimator is proposed using this knowledge. Third, we establish an asymptotically normal distribution for this corrected estimator under the high-dimensional setting. The second contribution of the paper concerns the correction of the likelihood-ratio statistic of the goodness-of- fit test to make it adapted to high-dimensional observations. The corrected statistic is proved asymptotically normal. Throughout the paper, Monte-Carlo experiments are conducted to assess the nite-sample behaviour of the methods. An application to returns of S&P 500 stock prices is also proposed.
34731652
On the sphericity test with large-dimensional observations
In this paper, we propose corrections to the likelihood ratio test and John's test for sphericity in large-dimensions. New formulas for the limiting parameters in the CLT for linear spectral statistics of sample covariance matrices with general fourth moments are first established. Using these formulas, we derive the asymptotic distribution of the two proposed test statistics under the null. These asymptotics are valid for general population, i.e. not necessarily Gaussian, provided a finite fourth-moment. Extensive Monte-Carlo experiments are conducted to assess the quality of these tests with a comparison to several existing methods from the literature. Moreover, we also obtain their asymptotic power functions under the alternative of a spiked population model as a specific alternative.
17588126
Signal detection in high dimension: The multispiked case
This paper deals with the local asymptotic structure, in the sense ofLe Cam’s asymptotic theory of statistical experiments, of the signal detectionproblem in high dimension. More precisely, we consider the problemof testing the null hypothesis of sphericity of a high-dimensional covariancematrix against an alternative of (unspecified) multiple symmetry-breakingdirections (multispiked alternatives). Simple analytical expressions for theasymptotic power envelope and the asymptotic powers of previously proposedtests are derived. These asymptotic powers are shown to lie verysubstantially below the envelope, at least for relatively small values of thenumber of symmetry-breaking directions under the alternative. In contrast,the asymptotic power of the likelihood ratio test based on the eigenvalues ofthe sample covariance matrix is shown to be close to that envelope. Theseresults extend to the case of multispiked alternatives the findings of an earlierstudy (Onatski, Moreira and Hallin, 2011) of the single-spiked case. The methods we are using here, however, are entirely new, as the Laplace approximationsconsidered in the single-spiked context do not extend to themultispiked case.
10491352
The efficient evaluation of the hypergeometric function of a matrix argument
We present new algorithms that efficiently approximate the hypergeometric function of a matrix argument through its expansion as a series of Jack functions. Our algorithms exploit the combinatorial properties of the Jack function, and have complexity that is only linear in the size of the matrix.
12038510
Joint density of eigenvalues in spiked multivariate models
The classical methods of multivariate analysis are based on the eigenvalues of one or two sample covariance matrices. In many applications of these methods, for example to high dimensional data, it is natural to consider alternative hypotheses which are a low rank departure from the null hypothesis. For rank one alternatives, this note provides a representation for the joint eigenvalue density in terms of a single contour integral. This will be of use for deriving approximate distributions for likelihood ratios and 'linear' statistics used in testing.
122648908
Symmetric functions and Hall polynomials
I. Symmetric functions II. Hall polynomials III. HallLittlewood symmetric functions IV. The characters of GLn over a finite field V. The Hecke ring of GLn over a finite field VI. Symmetric functions with two parameters VII. Zonal polynomials
1805465
MOPS: Multivariate orthogonal polynomials (symbolically)
In this paper we present a Maple library (MOPS) for computing Jack, Hermite, Laguerre, and Jacobi multivariate polynomials, as well as eigenvalue statistics for the Hermite, Laguerre, and Jacobi ensembles of random matrix theory. We also compute multivariate hypergeometric functions, and offer both symbolic and numerical evaluations for all these quantities. We prove that all algorithms are well-defined, analyze their complexity, and illustrate their performance in practice. Finally, we present a few applications of this library.
119133595
The beta-Wishart ensemble
We introduce a “broken-arrow” matrix model for the β-Wishart ensemble, which improves on the traditional bidiagonal model by generalizing to non-identity covariance parameters. We prove that its joint eigenvalue density involves the correct hypergeometric function of two matrix arguments, and a continuous parameter β > 0. If we choose β = 1, 2, 4, we recover the classical Wishart ensembles of general covariance over the reals, complexes, and quaternions. Jack polynomials are often defined as the eigenfunctions of the Laplace-Beltrami operator. We prove that Jack polynomials are in addition eigenfunctions of an integral operator defined as an average over a β-dependent measure on the sphere. When combined with an identity due to Stanley, we derive a definition of Jack polynomials. An efficient numerical algorithm is also presented for simulations. The algorithm makes use of secular equation software for broken arrow matrices currently unavailable in the popular technical computing languages. The simulation...
122583825
An Ising ferromagnet with discontinuous long-range order
An infinite one-dimensional Ising ferromagnetM with long-range interactions is constructed and proved to have the following properties. (1)M has an order-disorder phase transition at a finite temperature. (2) Any Ising ferromagnet of the same structure asM, but with interactions tending to zero with distance more rapidly than those ofM, cannot have a phase-transition. (3) The long-range-order parameter (thermal average of the spin-spin correlation at infinite distance) jumps discontinuously from zero in the disordered phase to a finite value in the ordered phase. All three properties have been conjectured by Anderson and Thouless to hold for a particular Ising ferromagnet which is relevant to the theory of the Kondo effect. AlthoughM is not identical to Anderson's model, the results proved forM support the validity of the physical arguments of Anderson and Thouless.
121400986
Asymptotics and Special Functions
A classic reference, intended for graduate students mathematicians, physicists, and engineers, this book can be used both as the basis for instructional courses and as a reference tool.