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.
  We develop a general approach to valid inference after model selection. In a nutshell, our approach produces post-selection inferences with the same frequency guarantees as those given by data splitting but are more powerful. At the core of our framework is a result that characterizes the distribution of a post-selection estimator conditioned on the selection event. We specialize the approach to model selection by the lasso to form valid condence intervals for the selected coecients and test whether all relevant variables have been included in the model. ABSTRACTThe article considers statistical inference for trends of high-dimensional time series. Based on a modified L2 distance between parametric and nonparametric trend estimators, we propose a d... A simple statistic is proposed for testing the complete independence of random variables having a multivariate normal distribution. The asymptotic null distribution of this statistic, as both the sample size and the number of variables go to infinity, is shown to be normal. Consequently, this test can be used when the number of variables is not small relative to the sample size and, in particular, even when the number of variables exceeds the sample size. The finite sample size performance of the normal approximation is evaluated in a simulation study and the results are compared to those of the likelihood ratio test. Copyright 2005, Oxford University Press.    In this article, we consider the problem of testing that the mean vector @m=0 in the model x"[email protected]+Cz"j,j=1,...,N, where z"j are random p-vectors, z"j=(z"i"j,...,z"p"j)^' and z"i"j are independently and identically distributed with finite four moments, i=1,...,p,j=1,...,N; that is x"i need not be normally distributed. We shall assume that C is a pxp non-singular matrix, and there are fewer observations than the dimension, [email protected]?p. We consider the test statistic T=[[email protected]?^'D"s^-^[email protected]?-np (n-2)] [2trR^2-p^2 n]^1^2, where [email protected]? is the sample mean vector, S=(s"i"j) is the sample covariance matrix, D"S= diag (s"1"1,...,s"p"p),R=D"s^-^1^2SD"s^-^1^2 and n=N-1. The asymptotic null and non-null distributions of the test statistic T are derived.  In recent years, there has been considerable theoretical development regarding variable selection consistency of penalized regression techniques, such as the lasso. However, there has been relatively little work on quantifying the uncertainty in these selection procedures. In this paper, we propose a new method for inference in high dimensions using a score test based on penalized regression. In this test, we perform penalized regression of an outcome on all but a single feature, and test for correlation of the residuals with the held-out feature. This procedure is applied to each feature in turn. Interestingly, when an @math penalty is used, the sparsity pattern of the lasso corresponds exactly to a decision based on the proposed test. Further, when an @math penalty is used, the test corresponds precisely to a score test in a mixed effects model, in which the effects of all but one feature are assumed to be random. We formulate the hypothesis being tested as a compromise between the null hypotheses tested in simple linear regression on each feature and in multiple linear regression on all features, and develop reference distributions for some well-known penalties. We also examine the behavior of the test on real and simulated data.  