Tests for high-dimensional covariance matrices

2019 ◽  
Vol 09 (03) ◽  
pp. 2050009
Author(s):  
Jing Chen ◽  
Xiaoyi Wang ◽  
Shurong Zheng ◽  
Baisen Liu ◽  
Ning-Zhong Shi
Keyword(s):  
Sample Size ◽  
Matrix Theory ◽  
Covariance Matrices ◽  
Asymptotic Distributions ◽  
High Dimensional ◽  
Frobenius Norm ◽  
Ratio Test ◽  
Finite Sample ◽  
Test Statistic ◽  
Sample Case

In this paper, we propose some new tests for high-dimensional covariance matrices that are applicable to generally distributed populations with finite fourth moments. The proposed test statistics are the maximum of the likelihood ratio test statistic and the statistic based on the Frobenius norm. The advantage of the new tests is the good performance in terms of power for both the traditional case, in which the dimension is much smaller than the sample size, and the high-dimensional case, in which the dimension is large compared to the sample size. In the one-sample case, the new test is proposed for testing the hypothesis that the high-dimensional covariance matrix equals an identity matrix. In the two-sample case, the new test is developed for testing the equality of two high-dimensional covariance matrices. By using the random matrix theory, the asymptotic distributions of the proposed new tests are derived under the assumption that the dimension and the sample size proportionally tend toward infinity. Finally, numerical studies are conducted to investigate the finite sample performance of the proposed new tests.

scholarly journals Monitoring Persistence Change in Heavy-Tailed Observations

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 936
Author(s):  
Dan Wang
Keyword(s):  
Kernel Method ◽  
Alternative Hypothesis ◽  
Null Distribution ◽  
Real Data ◽  
Ratio Test ◽  
Finite Sample ◽  
Test Statistic ◽  
Bootstrap Approximation ◽  
Heavy Tailed ◽  
Better Than

In this paper, a ratio test based on bootstrap approximation is proposed to detect the persistence change in heavy-tailed observations. This paper focuses on the symmetry testing problems of I(1)-to-I(0) and I(0)-to-I(1). On the basis of residual CUSUM, the test statistic is constructed in a ratio form. I prove the null distribution of the test statistic. The consistency under alternative hypothesis is also discussed. However, the null distribution of the test statistic contains an unknown tail index. To address this challenge, I present a bootstrap approximation method for determining the rejection region of this test. Simulation studies of artificial data are conducted to assess the finite sample performance, which shows that our method is better than the kernel method in all listed cases. The analysis of real data also demonstrates the excellent performance of this method.

scholarly journals Limit Properties of the Largest Entries of High-Dimensional Sample Covariance and Correlation Matrices

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Xue Ding
Keyword(s):  
Sample Size ◽  
Covariance Matrices ◽  
High Dimensional ◽  
Extreme Statistics ◽  
Correlation Matrices ◽  
Underlying Distribution ◽  
Sample Covariance Matrices ◽  
Sample Covariance ◽  
Sample Correlation

In this paper, we consider the limit properties of the largest entries of sample covariance matrices and the sample correlation matrices. In order to make the statistics based on the largest entries of the sample covariance matrices and the sample correlation matrices more applicable in high-dimensional tests, the identically distributed assumption of population is removed. Under some moment’s assumption of the underlying distribution, we obtain that the almost surely limit and asymptotical distribution of the extreme statistics as both the dimension p and sample size n tend to infinity.

TESTING INSTABILITY IN A PREDICTIVE REGRESSION MODEL WITH NONSTATIONARY REGRESSORS

2014 ◽  
Vol 31 (5) ◽  
pp. 953-980 ◽  
Author(s):  
Zongwu Cai ◽  
Yunfei Wang ◽  
Yonggang Wang
Keyword(s):  
Asymptotic Distributions ◽  
Monte Carlo Experiment ◽  
Finite Sample ◽  
Test Statistic ◽  
Predictive Regression ◽  
Testing Method ◽  
Type Test ◽  
Coefficient Vector ◽  
Alternative Hypotheses ◽  
The Stability

It is well known that allowing the coefficients to be time-varying in a predictive model with possibly nonstationary regressors can help to deal with instability in predictability associated with linear predictive models. In this paper, an L2-type test statistic is proposed to test the stability of the coefficient vector, and the asymptotic distributions of the proposed test statistic are developed under both null and alternative hypotheses. A Monte Carlo experiment is conducted to evaluate the finite sample performance of the proposed test statistic and an empirical example is examined to demonstrate the practical application of the proposed testing method.

scholarly journals Homogeneity tests of covariance matrices with high-dimensional longitudinal data

Biometrika ◽  
2019 ◽  
Vol 106 (3) ◽  
pp. 619-634 ◽  
Author(s):  
Ping-Shou Zhong ◽  
Runze Li ◽  
Shawn Santo
Keyword(s):  
Longitudinal Data ◽  
Sample Size ◽  
Time Course ◽  
Signal To Noise Ratio ◽  
Microarray Dataset ◽  
Gene Interaction ◽  
Repeated Measurements ◽  
High Dimensional ◽  
Test Statistic ◽  
Empirical Size

Summary This paper deals with the detection and identification of changepoints among covariances of high-dimensional longitudinal data, where the number of features is greater than both the sample size and the number of repeated measurements. The proposed methods are applicable under general temporal-spatial dependence. A new test statistic is introduced for changepoint detection, and its asymptotic distribution is established. If a changepoint is detected, an estimate of the location is provided. The rate of convergence of the estimator is shown to depend on the data dimension, sample size, and signal-to-noise ratio. Binary segmentation is used to estimate the locations of possibly multiple changepoints, and the corresponding estimator is shown to be consistent under mild conditions. Simulation studies provide the empirical size and power of the proposed test and the accuracy of the changepoint estimator. An application to a time-course microarray dataset identifies gene sets with significant gene interaction changes over time.

scholarly journals Efficient computation of limit spectra of sample covariance matrices

2015 ◽  
Vol 04 (04) ◽  
pp. 1550019 ◽  
Author(s):  
Edgar Dobriban
Keyword(s):  
Covariance Matrix ◽  
Spectral Distribution ◽  
Matrix Theory ◽  
Covariance Matrices ◽  
Discrete Distributions ◽  
High Dimensional ◽  
Efficient Computation ◽  
Usual Model ◽  
Sample Covariance ◽  
Distribution Of Eigenvalues

Models from random matrix theory (RMT) are increasingly used to gain insights into the behavior of statistical methods under high-dimensional asymptotics. However, the applicability of the framework is limited by numerical problems. Consider the usual model of multivariate statistics where the data is a sample from a multivariate distribution with a given covariance matrix. Under high-dimensional asymptotics, there is a deterministic map from the distribution of eigenvalues of the population covariance matrix (the population spectral distribution or PSD), to the of empirical spectral distribution (ESD). The current methods for computing this map are inefficient, and this limits the applicability of the theory. We propose a new method to compute numerically the ESD from an arbitrary input PSD. Our method, called SPECTRODE, finds the support and the density of the ESD to high precision; we prove this for finite discrete distributions. In computational experiments SPECTRODE outperforms existing methods by orders of magnitude in speed and accuracy. We apply it to compute expectations and contour integrals of the ESD, which are often central in applications. We also illustrate that SPECTRODE is directly useful in statistical problems, such as estimation and hypothesis testing for covariance matrices. Our proposal, implemented in open source software, may broaden the use of RMT in high-dimensional data analysis.

scholarly journals Ensemble-based estimates of eigenvector error for empirical covariance matrices

2018 ◽  
Vol 8 (2) ◽  
pp. 289-312
Author(s):  
Dane Taylor ◽  
Juan G Restrepo ◽  
François G Meyer
Keyword(s):  
Sample Size ◽  
Covariance Matrix ◽  
Covariance Matrices ◽  
Finite Sample ◽  
Large Matrix ◽  
The Matrix ◽  
Negligible Probability ◽  
Finite Sample Size ◽  
Matrix Ensemble ◽  
Empirical Covariance

Abstract Covariance matrices are fundamental to the analysis and forecast of economic, physical and biological systems. Although the eigenvalues $\{\lambda _i\}$ and eigenvectors $\{\boldsymbol{u}_i\}$ of a covariance matrix are central to such endeavours, in practice one must inevitably approximate the covariance matrix based on data with finite sample size $n$ to obtain empirical eigenvalues $\{\tilde{\lambda }_i\}$ and eigenvectors $\{\tilde{\boldsymbol{u}}_i\}$, and therefore understanding the error so introduced is of central importance. We analyse eigenvector error $\|\boldsymbol{u}_i - \tilde{\boldsymbol{u}}_i \|^2$ while leveraging the assumption that the true covariance matrix having size $p$ is drawn from a matrix ensemble with known spectral properties—particularly, we assume the distribution of population eigenvalues weakly converges as $p\to \infty $ to a spectral density $\rho (\lambda )$ and that the spacing between population eigenvalues is similar to that for the Gaussian orthogonal ensemble. Our approach complements previous analyses of eigenvector error that require the full set of eigenvalues to be known, which can be computationally infeasible when $p$ is large. To provide a scalable approach for uncertainty quantification of eigenvector error, we consider a fixed eigenvalue $\lambda $ and approximate the distribution of the expected square error $r= \mathbb{E}\left [\| \boldsymbol{u}_i - \tilde{\boldsymbol{u}}_i \|^2\right ]$ across the matrix ensemble for all $\boldsymbol{u}_i$ associated with $\lambda _i=\lambda $. We find, for example, that for sufficiently large matrix size $p$ and sample size $n> p$, the probability density of $r$ scales as $1/nr^2$. This power-law scaling implies that the eigenvector error is extremely heterogeneous—even if $r$ is very small for most eigenvectors, it can be large for others with non-negligible probability. We support this and further results with numerical experiments.

A CONSISTENT NONPARAMETRIC TEST ON SEMIPARAMETRIC SMOOTH COEFFICIENT MODELS WITH INTEGRATED TIME SERIES

2015 ◽  
Vol 32 (4) ◽  
pp. 988-1022 ◽  
Author(s):  
Yiguo Sun ◽  
Zongwu Cai ◽  
Qi Li
Keyword(s):  
Time Series ◽  
Nonparametric Test ◽  
Asymptotic Distributions ◽  
Varying Coefficient Model ◽  
Finite Sample ◽  
Test Statistic ◽  
Smooth Coefficients ◽  
Alternative Hypotheses ◽  
Constant Coefficients ◽  
Integrated Time Series

In this paper, we propose a simple nonparametric test for testing the null hypothesis of constant coefficients against nonparametric smooth coefficients in a semiparametric varying coefficient model with integrated time series. We establish the asymptotic distributions of the proposed test statistic under both null and alternative hypotheses. Moreover, we derive a central limit theorem for a degenerate second order U-statistic, which contains a mixture of stationary and nonstationary variables and is weighted locally on a stationary variable. This result is of independent interest and useful in other applications. Monte Carlo simulations are conducted to examine the finite sample performance of the proposed test.

scholarly journals Bartlett-corrected tests for varying precision beta regressions with application to environmental biometrics

PLoS ONE ◽  
2021 ◽  
Vol 16 (6) ◽  
pp. e0253349
Author(s):  
Ana C. Guedes ◽  
Francisco Cribari-Neto ◽  
Patrícia L. Espinheira
Keyword(s):  
Sample Size ◽  
Likelihood Ratio ◽  
Likelihood Ratio Test ◽  
Superior Performance ◽  
Model Parameters ◽  
Ratio Test ◽  
Test Statistics ◽  
Finite Sample ◽  
Alternative Approach ◽  
The Mean

Beta regressions are commonly used with responses that assume values in the standard unit interval, such as rates, proportions and concentration indices. Hypothesis testing inferences on the model parameters are typically performed using the likelihood ratio test. It delivers accurate inferences when the sample size is large, but can otherwise lead to unreliable conclusions. It is thus important to develop alternative tests with superior finite sample behavior. We derive the Bartlett correction to the likelihood ratio test under the more general formulation of the beta regression model, i.e. under varying precision. The model contains two submodels, one for the mean response and a separate one for the precision parameter. Our interest lies in performing testing inferences on the parameters that index both submodels. We use three Bartlett-corrected likelihood ratio test statistics that are expected to yield superior performance when the sample size is small. We present Monte Carlo simulation evidence on the finite sample behavior of the Bartlett-corrected tests relative to the standard likelihood ratio test and to two improved tests that are based on an alternative approach. The numerical evidence shows that one of the Bartlett-corrected typically delivers accurate inferences even when the sample is quite small. An empirical application related to behavioral biometrics is presented and discussed.

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