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 literature on inference for high-dimensional VAR models, most work focuses on the estimation problem. Song and Bickel ( @cite_6 ) investigate penalized least squares algorithms for different penalties, with some externally imposed assumptions on the temporal dependence. Theoretical guarantees on Dantzig type and Lasso type estimators are studied in @cite_4 and @cite_3 , but with Gaussian noise. Barigozzi and Brownlees ( @cite_8 ) consider the inference for stationary dependence structure built among variables, other than the parameters in the VAR model. In our work, we control the error bounds of Lasso and Dantzig type estimators for parameter matrices, with sub-Gaussian noise. Then we establish asymptotic distribution of test statistic based on this.
@cite_3 @cite_4 @cite_6 @cite_8
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We consider the problem of learning a coefficient vector xο ∈ RN from noisy linear observation y = Axo + ∈ Rn. In many contexts (ranging from model selection to image processing), it is desirable to construct a sparse estimator x. In this case, a popular approach consists in solving an l1-penalized least-squares problem known as the LASSO or basis pursuit denoising. For sequences of matrices A of increasing dimensions, with independent Gaussian entries, we prove that the normalized risk of the LASSO converges to a limit, and we obtain an explicit expression for this limit. Our result is the first rigorous derivation of an explicit formula for the asymptotic mean square error of the LASSO for random instances. The proof technique is based on the analysis of AMP, a recently developed efficient algorithm, that is inspired from graphical model ideas. Simulations on real data matrices suggest that our results can be relevant in a broad array of practical applications. The vector autoregressive (VAR) model is a powerful tool in learning complex time series and has been exploited in many fields. The VAR model poses some unique challenges to researchers: On one hand, the dimensionality, introduced by incorporating multiple numbers of time series and adding the order of the vector autoregression, is usually much higher than the time series length; On the other hand, the temporal dependence structure naturally present in the VAR model gives rise to extra difficulties in data analysis. The regular way in cracking the VAR model is via "least squares" and usually involves adding different penalty terms (e.g., ridge or lasso penalty) in handling high dimensionality. In this manuscript, we propose an alternative way in estimating the VAR model. The main idea is, via exploiting the temporal dependence structure, formulating the estimating problem to a linear program. There is instant advantage of the proposed approach over the lasso-type estimators: The estimation equation can be decomposed to multiple sub-equations and accordingly can be solved efficiently using parallel computing. Besides that, we also bring new theoretical insights into the VAR model analysis. So far the theoretical results developed in high dimensions (e.g., Song and Bickel, 2011 and Kock and Callot, 2015) are based on stringent assumptions that are not transparent. Our results, on the other hand, show that the spectral norms of the transition matrices play an important role in estimation accuracy and build estimation and prediction consistency accordingly. Moreover, we provide some experiments on both synthetic and real-world equity data. We show that there are empirical advantages of our method over the lasso-type estimators in parameter estimation and forecasting. We study the fundamental problems of variance and risk estimation in high dimensional statistical modeling. In particular, we consider the problem of learning a coefficient vector Theta 0 is an element of Rp from noisy linear observations y = X Theta 0 + w is an element of Rn (p > n) and the popular estimation procedure of solving the '1-penalized least squares objective known as the LASSO or Basis Pursuit DeNoising (BPDN). In this context, we develop new estimators for the '2 estimation risk k Theta b- Theta 0k2 and the variance of the noise when distributions of Theta 0 and w are unknown. These can be used to select the regularization parameter optimally. Our approach combines Stein's unbiased risk estimate [Ste81] and the recent results of [BM12a] [BM12b] on the analysis of approximate message passing and the risk of LASSO. We establish high-dimensional consistency of our estimators for sequences of matrices X of increasing dimensions, with independent Gaussian entries. We establish validity for a broader class of Gaussian designs, conditional on a certain conjecture from statistical physics. To the best of our knowledge, this result is the first that provides an asymptotically consistent risk estimator for the LASSO solely based on data. In addition, we demonstrate through simulations that our variance estimation outperforms several existing methods in the literature. Given @math noisy samples with @math dimensions, where @math , we show that the multi-step thresholding procedure based on the Lasso -- we call it the Thresholded Lasso , can accurately estimate a sparse vector @math in a linear model @math , where @math is a design matrix normalized to have column @math norm @math , and @math . We show that under the restricted eigenvalue (RE) condition (Bickel-Ritov-Tsybakov 09), it is possible to achieve the @math loss within a logarithmic factor of the ideal mean square error one would achieve with an oracle while selecting a sufficiently sparse model -- hence achieving sparse oracle inequalities ; the oracle would supply perfect information about which coordinates are non-zero and which are above the noise level. In some sense, the Thresholded Lasso recovers the choices that would have been made by the @math penalized least squares estimators, in that it selects a sufficiently sparse model without sacrificing the accuracy in estimating @math and in predicting @math . We also show for the Gauss-Dantzig selector (Cand e s-Tao 07), if @math obeys a uniform uncertainty principle and if the true parameter is sufficiently sparse, one will achieve the sparse oracle inequalities as above, while allowing at most @math irrelevant variables in the model in the worst case, where @math is the smallest integer such that for @math , @math . Our simulation results on the Thresholded Lasso match our theoretical analysis excellently.