Finite Fourth Moment Research Articles

ON FINITE RANK DEFORMATIONS OF WIGNER MATRICES II: DELOCALIZED PERTURBATIONS

We study the distribution of the outliers in the spectrum of finite rank deformations of Wigner random matrices. We assume that the matrix entries have finite fourth moment and extend the results by Capitaine, Donati-Martin, and Féral for perturbations whose eigenvectors are delocalized.

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.

Fluctuations of Matrix Entries of Regular Functions of Wigner Matrices

We study the fluctuations of the matrix entries of regular functions of Wigner random matrices in the limit when the matrix size goes to infinity. In the case of the Gaussian ensembles (GOE and GUE) this problem was considered by A.Lytova and L.Pastur in J. Stat. Phys., v.134, 147-159 (2009). Our results are valid provided the off-diagonal matrix entries have finite fourth moment, the diagonal matrix entries have finite second moment, and the test functions have four continuous derivatives in a neighborhood of the support of the Wigner semicircle law.

Asymptotic properties of eigenmatrices of a large sample covariance matrix

Let Sn = 1/n XnXn* where Xn = {Xij} is a p × n matrix with i.i.d. complex standardized entries having finite fourth moments. Let $Y_{n}(\mathbf{t}_{1},\mathbf{t}_{2},\sigma)=\sqrt{p}({\mathbf{x}}_{n}(\mathbf{t}_{1})^{*}(S_{n}+\sigma I)^{-1}{\mathbf{x}}_{n}(\mathbf{t}_{2})-{\mathbf{x}}_{n}(\mathbf{t}_{1})^{*}{\mathbf{x}}_{n}(\mathbf{t}_{2})m_{n}(\sigma))$ in which σ > 0 and mn(σ) = ∫d Fyn(x)/(x + σ) where Fyn(x) is the Marčenko–Pastur law with parameter yn = p/n; which converges to a positive constant as n → ∞, and xn(t1) and xn(t2) are unit vectors in ${\mathbb{C}}^{p}$, having indices t1 and t2, ranging in a compact subset of a finite-dimensional Euclidean space. In this paper, we prove that the sequence Yn(t1, t2, σ) converges weakly to a (2m + 1)-dimensional Gaussian process. This result provides further evidence in support of the conjecture that the distribution of the eigenmatrix of Sn is asymptotically close to that of a Haar-distributed unitary matrix.

Limiting spectral distributions of some band matrices

We use the method of moments to establish the limiting spectral distribution (LSD) of appropriately scaled large dimensional random symmetric circulant, reverse circulant, Toeplitz and Hankel matrices which have suitable band structures. The input sequence used to construct these matrices is assumed to be either i.i.d. with mean zero and variance one or independent and appropriate finite fourth moment. The class of LSD includes the normal and the symmetrized square root of chi-square with two degrees of freedom. In several other cases, explicit forms of the limit do not seem to be obtainable but the limits can be shown to be symmetric and their second and the fourth moments can be calculated with some effort. Simulations suggest some further properties of the limits.

HIGHER-ORDER ACCURATE, POSITIVE SEMIDEFINITE ESTIMATION OF LARGE-SAMPLE COVARIANCE AND SPECTRAL DENSITY MATRICES

A new class of large-sample covariance and spectral density matrix estimators is proposed based on the notion of flat-top kernels. The new estimators are shown to be higher-order accurate when higher-order accuracy is possible. A discussion on kernel choice is presented as well as a supporting finite-sample simulation. The problem of spectral estimation under a potential lack of finite fourth moments is also addressed. The higher-order accuracy of flat-top kernel estimators typically comes at the sacrifice of the positive semidefinite property. Nevertheless, we show how a flat-top estimator can be modified to become positive semidefinite (even strictly positive definite) while maintaining its higher-order accuracy. In addition, an easy (and consistent) procedure for optimal bandwidth choice is given; this procedure estimates the optimal bandwidth associated with each individual element of the target matrix, automatically sensing (and adapting to) the underlying correlation structure.

Simultaneous critical values for $t$-tests in very high dimensions

This article considers the problem of multiple hypothesis testing using $t$-tests. The observed data are assumed to be independently generated conditional on an underlying and unknown two-state hidden model. We propose an asymptotically valid data-driven procedure to find critical values for rejection regions controlling the $k$-familywise error rate ($k$-FWER), false discovery rate (FDR) and the tail probability of false discovery proportion (FDTP) by using one-sample and two-sample $t$-statistics. We only require a finite fourth moment plus some very general conditions on the mean and variance of the population by virtue of the moderate deviations properties of $t$-statistics. A new consistent estimator for the proportion of alternative hypotheses is developed. Simulation studies support our theoretical results and demonstrate that the power of a multiple testing procedure can be substantially improved by using critical values directly, as opposed to the conventional $p$-value approach. Our method is applied in an analysis of the microarray data from a leukemia cancer study that involves testing a large number of hypotheses simultaneously.

Estimation and Asymptotic Inference in the AR-ARCH Model

This article studies asymptotic properties of the quasi-maximum likelihood estimator (QMLE) for the parameters in the autoregressive (AR) model with autoregressive conditional heteroskedastic (ARCH) errors. A modified QMLE (MQMLE) is also studied. This estimator is based on truncation of individual terms of the likelihood function and is related to the recent so-called self-weighted QMLE in Ling (2007b). We show that the MQMLE is asymptotically normal irrespectively of the existence of finite moments, as geometric ergodicity alone suffice. Moreover, our included simulations show that the MQMLE is remarkably well-behaved in small samples. On the other hand, the ordinary QMLE, as is well-known, requires finite fourth order moments for asymptotic normality. But based on our considerations and simulations, we conjecture that in fact only geometric ergodicity and finite second order moments are needed for the QMLE to be asymptotically normal. Finally, geometric ergodicity for AR-ARCH processes is shown to hold under mild and classic conditions on the AR and ARCH processes.

How Close is the Sample Covariance Matrix to the Actual Covariance Matrix?

Given a probability distribution in ℝ n with general (nonwhite) covariance, a classical estimator of the covariance matrix is the sample covariance matrix obtained from a sample of N independent points. What is the optimal sample size N=N(n) that guarantees estimation with a fixed accuracy in the operator norm? Suppose that the distribution is supported in a centered Euclidean ball of radius $O(\sqrt{n})$ . We conjecture that the optimal sample size is N=O(n) for all distributions with finite fourth moment, and we prove this up to an iterated logarithmic factor. This problem is motivated by the optimal theorem of Rudelson (J. Funct. Anal. 164:60–72, 1999), which states that N=O(nlog n) for distributions with finite second moment, and a recent result of Adamczak et al. (J. Am. Math. Soc. 234:535–561, 2010), which guarantees that N=O(n) for subexponential distributions.

Optimal Scaling of Random Walk Metropolis Algorithms with Non-Gaussian Proposals

The asymptotic optimal scaling of random walk Metropolis (RWM) algorithms with Gaussian proposal distributions is well understood for certain specific classes of target distributions. These asymptotic results easily extend to any light tailed proposal distribution with finite fourth moment. However, heavy tailed proposal distributions such as the Cauchy distribution are known to have a number of desirable properties, and in many situations are preferable to light tailed proposal distributions. Therefore we consider the question of scaling for Cauchy distributed proposals for a wide range of independent and identically distributed (iid) product densities. The results are somewhat surprising as to when and when not Cauchy distributed proposals are preferable to Gaussian proposal distributions. This provides motivation for finding proposal distributions which improve on both Gaussian and Cauchy proposals, both for finite dimensional target distributions and asymptotically as the dimension of the target density, d → ∞. Throughout we seek the scaling of the proposal distribution which maximizes the expected squared jumping distance (ESJD).

A Note on Simultaneous Confidence Intervals for Ratio of Variances

An explicit formula for confidence intervals for ratios of variances of several populations is presented. The intervals are based on jackknife statistics and the critical point of the studentized range distribution. The asymptotic probability of coverage is not less than the nominal value provided that the distributions of the sampled populations belong to a location-scale family of probabilities with finite fourth moment.

Nonparametric estimation of the volatility function in a high-frequency model corrupted by noise

We consider the models Yi,n=∫0i/nσ(s)dWs+τ(i/n)ei,n, and Yi,n=σ(i/n)Wi/n+τ(i/n)ei,n, i=1,…,n, where (Wt)t∈[0,1] denotes a standard Brownian motion and ei,n are centered i.i.d. random variables with E (ei,n2)=1 and finite fourth moment. Furthermore, σ and τ are unknown deterministic functions and (Wt)t∈[0,1] and (e1,n,…,en,n) are assumed to be independent processes. Based on a spectral decomposition of the covariance structures we derive series estimators for σ2 and τ2 and investigate their rate of convergence of the MISE in dependence of their smoothness. To this end specific basis functions and their corresponding Sobolev ellipsoids are introduced and we show that our estimators are optimal in minimax sense. Our work is motivated by microstructure noise models. A major finding is that the microstructure noise ei,n introduces an additionally degree of ill-posedness of 1/2; irrespectively of the tail behavior of ei,n. The performance of the estimates is illustrated by a small numerical study.

Almost Sure Limit of the Smallest Eigenvalue of Some Sample Correlation Matrices

Let X(n)=(Xij) be a p×n data matrix, where the n columns form a random sample of size n from a certain p-dimensional distribution. Let R(n)=(ρij) be the p×p sample correlation coefficient matrix of X(n), and \(S^{(n)}=(1/n)X^{(n)}(X^{(n)})^{\ast}-\bar{X}\bar{X}^{\ast}\) be the sample covariance matrix of X(n), where \(\bar{X}\) is the mean vector of the n observations. Assuming that Xij are independent and identically distributed with finite fourth moment, we show that the smallest eigenvalue of R(n) converges almost surely to the limit \((1-\sqrt{c}\,)^{2}\) as n→∞ and p/n→c∈(0,∞). We accomplish this by showing that the smallest eigenvalue of S(n) converges almost surely to \((1-\sqrt{c}\,)^{2}\) .

A test of the hypothesis of partial common principal components

Abstract A test of the equality of the first h eigenvectors of covariance matrices of several populations is constructed without the assumption that the sampled distributions are Gaussian. It is proved that the test statistic is asymptotically chi-square distributed. In this general setting, an explicit formula for column space of the asymptotic covariance matrix of the sample eigenvectors is derived and the rank of this matrix is computed. An essential assumption in deriving the asymptotic distribution of the presented test statistic is the existence of the finite fourth moments and the simplicity of the h largest eigenvalues of population covariance matrices, which makes possible to use the formulas for derivatives of eigenvectors of symmetric matrices.

Circular law, extreme singular values and potential theory

Consider the empirical spectral distribution of complex random n × n matrix whose entries are independent and identically distributed random variables with mean zero and variance 1 / n . In this paper, via applying potential theory in the complex plane and analyzing extreme singular values, we prove that this distribution converges, with probability one, to the uniform distribution over the unit disk in the complex plane, i.e. the well known circular law, under the finite fourth moment assumption on matrix elements.

A Constant on a Uniform Bound of a Combinatorial Central Limit Theorem

Let n be a positive integer and Y(i, j), i, j = 1, ..., n, be random variables with finite fourth moments. Let ? be a randompermutation on {1, ..., n} which independent of Y(i, j)’s. In this paper, we use Stein’s method and the technique from(Laipaporn, K., 2008) to give a uniform bound in a combinatorial central limit theorem of W =n *i=1Y(i, ?(i)). For asufficient large n, we yield the rate27.72?n. This constant is better than the result in (Neammanee, K., 2005).Keywords: Uniform bound, Combinatorial central limit theorem, Stein’s method, Random permutation

A Robust Score Test for Testing Several Coefficients of Variation with Unknown Underlying Distributions

A parametric robust test is proposed for comparing several coefficients of variation. This test is derived by properly correcting the normal likelihood function according to the technique suggested by Royall and Tsou. The proposed test statistic is asymptotically valid for general random variables, as long as their underlying distributions have finite fourth moments. Simulation studies and real data analyses are provided to demonstrate the effectiveness of the novel robust procedure.

Reaching the best possible rate of convergence to equilibrium for solutions of Kac’s equation via central limit theorem

Let f(⋅, t) be the probability density function which represents the solution of Kac’s equation at time t, with initial data f0, and let gσ be the Gaussian density with zero mean and variance σ2, σ2 being the value of the second moment of f0. This is the first study which proves that the total variation distance between f(⋅, t) and gσ goes to zero, as t→+∞, with an exponential rate equal to −1/4. In the present paper, this fact is proved on the sole assumption that f0 has finite fourth moment and its Fourier transform ϕ0 satisfies |ϕ0(ξ)|=o(|ξ|−p) as |ξ|→+∞, for some p>0. These hypotheses are definitely weaker than those considered so far in the state-of-the-art literature, which in any case, obtains less precise rates.

Large deviations for eigenvalues of sample covariance matrices, with applications to mobile communication systems

We study sample covariance matrices of the form W = (1 / n)CCT, where C is a k x n matrix with independent and identically distributed (i.i.d.) mean 0 entries. This is a generalization of the so-called Wishart matrices, where the entries of C are i.i.d. standard normal random variables. Such matrices arise in statistics as sample covariance matrices, and the high-dimensional case, when k is large, arises in the analysis of DNA experiments. We investigate the large deviation properties of the largest and smallest eigenvalues of W when either k is fixed and n → ∞ or kn → ∞ with kn = o(n / log log n), in the case where the squares of the i.i.d. entries have finite exponential moments. Previous results, proving almost sure limits of the eigenvalues, require only finite fourth moments. Our most explicit results for large k are for the case where the entries of C are ∓ 1 with equal probability. We relate the large deviation rate functions of the smallest and largest eigenvalues to the rate functions for i.i.d. standard normal entries of C. This case is of particular interest since it is related to the problem of decoding of a signal in a code-division multiple-access (CDMA) system arising in mobile communication systems. In this example, k is the number of users in the system and n is the length of the coding sequence of each of the users. Each user transmits at the same time and uses the same frequency; the codes are used to distinguish the signals of the separate users. The results imply large deviation bounds for the probability of a bit error due to the interference of the various users.

Large deviations for eigenvalues of sample covariance matrices, with applications to mobile communication systems

We study sample covariance matrices of the form W = (1 / n) C C T, where C is a k x n matrix with independent and identically distributed (i.i.d.) mean 0 entries. This is a generalization of the so-called Wishart matrices, where the entries of C are i.i.d. standard normal random variables. Such matrices arise in statistics as sample covariance matrices, and the high-dimensional case, when k is large, arises in the analysis of DNA experiments. We investigate the large deviation properties of the largest and smallest eigenvalues of W when either k is fixed and n → ∞ or k n → ∞ with k n = o(n / log log n), in the case where the squares of the i.i.d. entries have finite exponential moments. Previous results, proving almost sure limits of the eigenvalues, require only finite fourth moments. Our most explicit results for large k are for the case where the entries of C are ∓ 1 with equal probability. We relate the large deviation rate functions of the smallest and largest eigenvalues to the rate functions for i.i.d. standard normal entries of C . This case is of particular interest since it is related to the problem of decoding of a signal in a code-division multiple-access (CDMA) system arising in mobile communication systems. In this example, k is the number of users in the system and n is the length of the coding sequence of each of the users. Each user transmits at the same time and uses the same frequency; the codes are used to distinguish the signals of the separate users. The results imply large deviation bounds for the probability of a bit error due to the interference of the various users.