arXiv subjects

Alexei Onatski

Publications and source records attributed to Alexei Onatski.

13 recordsLinked to original sources

arXiv2025

Precise asymptotics for the norm of large random rectangular Toeplitz matrices

We study the spectral norm of large rectangular random Toeplitz and circulant matrices with independent entries. For Toeplitz matrices, we show that the scaled norm converges to the norm of a bilinear operator defined via the pointwise product of two scaled sine kernel operators. In the square case, this limit reduces to the squared $2\!\to\!4$ norm of the sine kernel operator, in agreement with the result of Sen and Vir\'{a}g (2013). For $p\times n$ circulant matrices, we show that their norm divided by $\sqrt{p\log n}$ converges in probability to 1. We further investigate the finite-sample performance of these asymptotic results via Monte Carlo experiments, which reveal both non-negligible bias and dispersion. For circulant matrices, a higher-order asymptotic analysis in the Gaussian case explains these effects, connects the fluctuations to shifted Gumbel distributions, and suggests a natural conjecture on the limiting law.

math.PR↗

arXiv2024

The smallest singular value of large random rectangular Toeplitz and circulant matrices

Let $x_i$, $i\in\mathbb{Z}$ be a sequence of i.i.d. standard normal random variables. Consider rectangular Toeplitz $\mathbf{X}=\left(x_{j-i}\right)_{1\leq i\leq p,1\leq j\leq n}$ and circulant $\mathbf{X}=\left(x_{(j-i)\mod n}\right)_{1\leq i\leq p,1\leq j\leq n}$ matrices. Let $p,n\rightarrow\infty$ so that $p/n\rightarrow c\in(0,1]$. We prove that the smallest eigenvalue of $\frac{1}{n}\mathbf{X}\mathbf{X}^\top$ converges to zero in probability and in expectation. We establish a lower bound on the rate of this convergence. The lower bound is faster than any poly-log but slower than any polynomial rate. For the ``rectangular circulant'' matrices, we also establish a polynomial upper bound on the convergence rate, which is a simple explicit function of $c$.

math.PR↗

arXiv2024

Spin glass to paramagnetic transition and triple point in Spherical SK model

This paper studies spin glass to paramagnetic transition in the Spherical Sherrington-Kirkpatrick model with ferromagnetic Curie-Weiss interaction with coupling constant $J$ and inverse temperature $β$. The disorder of the system is represented by a general Wigner matrix. We confirm a conjecture of \cite{Baik2016} and \cite{Baik2017}, that the critical window of temperatures for this transition is $β= 1 + bN^{-1/3} \sqrt{\log N}$ with $b\in\mathbb{R}$. The limiting distribution of the scaled free energy is Gaussian for negative $b$ and a weighted linear combination of independent Gaussian and Tracy-Widom components for positive $b$. In the special case where the Wigner matrix is from the Gaussian Orthogonal or Unitary Ensemble, we describe the triple point transition between spin glass, paramagnetic, and ferromagnetic regimes in a critical window for $(β, J)$ around the triple point $(1,1)$: the Tracy-Widom component is replaced by the one parameter family of deformations described by Bloemendal and Virag, \cite{BloVirI}.

math.PR↗

arXiv2022

An edge CLT for the log determinant of Wigner ensembles

We derive a Central Limit Theorem (CLT) for $\log \left\vert\det \left( W_{N}-E_{N}\right)\right\vert,$ where $W_{N}$ is a Wigner matrix, and $E_{N}$ is local to the edge of the semi-circle law. Precisely, $E_N=2+N^{-2/3}σ_N$ with $σ_N$ being either a constant (possibly negative), or a sequence of positive real numbers, slowly diverging to infinity so that $σ_N \ll \log^{2} N$. We also extend our CLT to cover spiked Wigner matrices. Our interest in the CLT is motivated by its applications to statistical testing in critically spiked models and to the fluctuations of the free energy in the spherical Sherrington-Kirkpatrick model of statistical physics.

math.PR↗

arXiv2021

Set coverage and robust policy

When conducting inference on partially identified parameters, confidence regions may cover the whole identified set with a prescribed probability, to which we will refer as set coverage, or they may cover each of its point with a prescribed probability, to which we will refer as point coverage. Since set coverage implies point coverage, confidence regions satisfying point coverage are generally preferred on the grounds that they may be more informative. The object of this note is to describe a decision problem in which, contrary to received wisdom, point coverage is clearly undesirable.

econ.EM↗

arXiv2018

Testing in high-dimensional spiked models

We consider the five classes of multivariate statistical problems identified by James (1964), which together cover much of classical multivariate analysis, plus a simpler limiting case, symmetric matrix denoising. Each of James' problems involves the eigenvalues of $E^{-1}H$ where $H$ and $E$ are proportional to high dimensional Wishart matrices. Under the null hypothesis, both Wisharts are central with identity covariance. Under the alternative, the non-centrality or the covariance parameter of $H$ has a single eigenvalue, a spike, that stands alone. When the spike is smaller than a case-specific phase transition threshold, none of the sample eigenvalues separate from the bulk, making the testing problem challenging. Using a unified strategy for the six cases, we show that the log likelihood ratio processes parameterized by the value of the sub-critical spike converge to Gaussian processes with logarithmic correlation. We then derive asymptotic power envelopes for tests for the presence of a spike.

math.ST↗

arXiv2016

Alternative asymptotics for cointegration tests in large VARs

Johansen's (1988, 1991) likelihood ratio test for cointegration rank of a Gaussian VAR depends only on the squared sample canonical correlations between current changes and past levels of a simple transformation of the data. We study the asymptotic behavior of the empirical distribution of those squared canonical correlations when the number of observations and the dimensionality of the VAR diverge to infinity simultaneously and proportionally. We find that the distribution almost surely weakly converges to the so-called Wachter distribution. This finding provides a theoretical explanation for the observed tendency of Johansen's test to find "spurious cointegration". It also sheds light on the workings and limitations of the Bartlett correction approach to the over-rejection problem. We propose a simple graphical device, similar to the scree plot, for a preliminary assessment of cointegration in high-dimensional VARs.

math.ST↗

arXiv2014

Local Asymptotic Normality of the spectrum of high-dimensional spiked F-ratios

We consider two types of spiked multivariate F distributions: a scaled distribution with the scale matrix equal to a rank-one perturbation of the identity, and a distribution with trivial scale, but rank-one non-centrality. The norm of the rank-one matrix (spike) parameterizes the joint distribution of the eigenvalues of the corresponding F matrix. We show that, for a spike located above a phase transition threshold, the asymptotic behavior of the log ratio of the joint density of the eigenvalues of the F matrix to their joint density under a local deviation from this value depends only on the largest eigenvalue $λ_{1}$. Furthermore, $λ_{1}$ is asymptotically normal, and the statistical experiment of observing all the eigenvalues of the F matrix converges in the Le Cam sense to a Gaussian shift experiment that depends on the asymptotic mean and variance of $λ_{1}$. In particular, the best statistical inference about a sufficiently large spike in the local asymptotic regime is based on the largest eigenvalue only. As a by-product of our analysis, we establish joint asymptotic normality of a few of the largest eigenvalues of the multi-spiked F matrix when the corresponding spikes are above the phase transition threshold.

math.ST↗

arXiv2013

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 contiguous alternatives, 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, Ben Arous and Peche [Ann. Probab. 33 (2005) 1643-1697] the limiting process is degenerate, and discrimination between the null and the alternative is asymptotically certain. When the norm is below the threshold, the limiting process is nondegenerate, and 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 (i.e., equals the asymptotic size), whereas that of the eigenvalue-based likelihood ratio test is strictly larger than the size, and close to the power envelope.

math.ST↗

arXiv2012

Signal Detection in High Dimension: The Multispiked Case

This paper deals with the local asymptotic structure, in the sense of Le Cam's asymptotic theory of statistical experiments, of the signal detection problem in high dimension. More precisely, we consider the problem of testing the null hypothesis of sphericity of a high-dimensional covariance matrix against an alternative of (unspecified) multiple symmetry-breaking directions (\textit{multispiked} alternatives). Simple analytical expressions for the asymptotic power envelope and the asymptotic powers of previously proposed tests are derived. These asymptotic powers are shown to lie very substantially below the envelope, at least for relatively small values of the number of symmetry-breaking directions under the alternative. In contrast, the asymptotic power of the likelihood ratio test based on the eigenvalues of the sample covariance matrix is shown to be close to that envelope. These results extend to the case of multispiked alternatives the findings of an earlier study (Onatski, Moreira and Hallin, 2011) of the single-spiked case. The methods we are using here, however, are entirely new, as the Laplace approximations considered in the single-spiked context do not extend to the multispiked case.

math.ST↗

arXiv2012

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 $_{0}F_{0}^{(α)}$of two $p\times p$ matrix arguments as a repeated contour integral of the hypergeometric function $_{0}F_{0}^{(α)}$of two $r\times r$ matrix arguments.

math.ST↗

arXiv2008

The Tracy--Widom limit for the largest eigenvalues of singular complex Wishart matrices

This paper extends the work of El Karoui [Ann. Probab. 35 (2007) 663--714] which finds the Tracy--Widom limit for the largest eigenvalue of a nonsingular $p$-dimensional complex Wishart matrix $W_{\mathbb{C}}(Ω_p,n)$ to the case of several of the largest eigenvalues of the possibly singular $(n<p)$ matrix $W_{\mathbb{C}}(Ω_p,n).$ As a byproduct, we extend all results of Baik, Ben Arous and Peche [Ann. Probab. 33 (2005) 1643--1697] to the singular Wishart matrix case. We apply our findings to obtain a 95% confidence set for the number of common risk factors in excess stock returns.

math.PR↗

arXiv2004

Dynamics of Interest Rate Curve by Functional Auto-Regression

The paper uses functional auto-regression to predict the dynamics of interest rate curve. It estimates the auto-regressive operator by extending methods of the reduced-rank auto-regression to the functional data. Such an estimation technique is better suited for prediction purposes as opposed to the methods based either on principal components or canonical correlations. The consistency of the estimator is proved using methods of operator theory. The estimation method is used to analyze dynamics of Eurodollar futures rates. The results suggest that future movements of interest rates are predictable at 1-year horizons.

math.ST↗