High-dimensional auto-regressive models provide a natural way to model influence between @math actors given multi-variate time series data for @math time intervals. While there has been considerable work on network estimation, there is limited work in the context of inference and hypothesis testing. In particular, prior work on hypothesis testing in time series has been restricted to linear Gaussian auto-regressive models. From a practical perspective, it is important to determine suitable statistical tests for connections between actors that go beyond the Gaussian assumption. In the context of time series models, confidence intervals present additional estimators since most estimators such as the Lasso and Dantzig selectors are biased which has led to estimators. In this paper we address these challenges and provide convergence in distribution results and confidence intervals for the multi-variate AR(p) model with sub-Gaussian noise, a generalization of Gaussian noise that broadens applicability and presents numerous technical challenges. The main technical challenge lies in the fact that unlike Gaussian random vectors, for sub-Gaussian vectors zero correlation does not imply independence. The proof relies on using an intricate truncation argument to develop novel concentration bounds for quadratic forms of dependent sub-Gaussian random variables. Our convergence in distribution results hold provided @math , where @math and @math refer to sparsity parameters which matches existed results for hypothesis testing with i.i.d. samples. We validate our theoretical results with simulation results for both block-structured and chain-structured networks.
In the high-dimensional hypothesis testing literature, there is some work regarding to testing for high-dimensional mean vector ( @cite_34 ), covariance matrices ( @cite_10 , @cite_33 ) and independence among variables ( @cite_24 ). While for testing on regression parameters, most work assumes i.i.d samples. @cite_19 , @cite_12 and @cite_22 proposes methods to test whether a covariate should be selected conditioning on the selection of some other covariates. A penalized score test depending on the tuning parameter @math is considered in @cite_25 . Our work follows the a line of work by @cite_2 , @cite_14 , @cite_5 and @cite_11 , the de-sparsifying or decorrelated literature. We construct a VAR version of decorrelated score test proposed by @cite_11 . Chen and Wu ( @cite_20 ) tackles the hypothesis testing problem for time series data as well, but they are testing the trend in a time series, instead of the autoregressive parameter which encodes the influence structure among variables.
@cite_14 @cite_33 @cite_22 @cite_20 @cite_24 @cite_19 @cite_2 @cite_5 @cite_34 @cite_10 @cite_25 @cite_12 @cite_11
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In the high-dimensional setting, this article considers three interrelated problems: (a) testing the equality of two covariance matrices and ; (b) recovering the support of ; and (c) testing the equality of and row by row. We propose a new test for testing the hypothesis H 0: and investigate its theoretical and numerical properties. The limiting null distribution of the test statistic is derived and the power of the test is studied. The test is shown to enjoy certain optimality and to be especially powerful against sparse alternatives. The simulation results show that the test significantly outperforms the existing methods both in terms of size and power. Analysis of a prostate cancer dataset is carried out to demonstrate the application of the testing procedures. When the null hypothesis of equal covariance matrices is rejected, it is often of significant interest to further investigate how they differ from each other. Motivated by applications in genomics, we also consider recovering the support of and ... When faced with high-dimensional data, one often uses principal component analysis (PCA) for dimension reduction. Classical PCA constructs a set of uncorrelated variables, which correspond to eigenvectors of the sample covariance matrix. However, it is well-known that this covariance matrix is strongly affected by anomalous observations. It is therefore necessary to apply robust methods that are resistant to possible outliers. Li and Chen [J. Am. Stat. Assoc. 80 (1985) 759] proposed a solution based on projection pursuit (PP). The idea is to search for the direction in which the projected observations have the largest robust scale. In subsequent steps, each new direction is constrained to be orthogonal to all previous directions. This method is very well suited for high-dimensional data, even when the number of variables p is higher than the number of observations n. However, the algorithm of Li and Chen has a high computational cost. In the references [C. Croux, A. Ruiz-Gazen, in COMPSTAT: Proceedings in Computational Statistics 1996, Physica-Verlag, Heidelberg, 1996, pp. 211–217; C. Croux and A. Ruiz-Gazen, High Breakdown Estimators for Principal Components: the Projection-Pursuit Approach Revisited, 2000, submitted for publication.], a computationally much more attractive method is presented, but in high dimensions (large p) it has a numerical accuracy problem and still consumes much computation time. In this paper, we construct a faster two-step algorithm that is more stable numerically. The new algorithm is illustrated on a data set with four dimensions and on two chemometrical data sets with 1200 and 600 dimensions. The problem of testing the mean vector in a high-dimensional setting is considered. Up to date, most high-dimensional tests for the mean vector only make use of the marginal information from the variables, and do not incorporate the correlation information into the test statistics. A new testing procedure is proposed, which makes use of the covariance information between the variables. The new approach is novel in that it can select important variables that contain evidence against the null hypothesis and reduce the impact of noise accumulation. Simulations and real data analysis demonstrate that the new test has higher power than some competing methods proposed in the literature. This article is concerned with simultaneous tests on linear regression coefficients in high-dimensional settings. When the dimensionality is larger than the sample size, the classic @math -test is not applicable since the sample covariance matrix is not invertible. In order to overcome this issue, both Goeman, Finos and van Houwelingen (2011) and Zhong and Chen (2011) proposed their test procedures after excluding the @math term in @math -statistics. However, both these two test are not invariant under the group of scalar transformations. In order to treat those variables in a fair' way, we proposed a new test statistic and establish its asymptotically normal under certain mild conditions. Simulation studies showed that our test procedure performs very well in many cases.