In this article, we investigate the problem of simultaneous change point inference and structure recovery in the context of high dimensional Gaussian graphical models with possible abrupt changes. In particular, motivated by neighborhood selection, we incorporate a threshold variable and an unknown threshold parameter into a joint sparse regression model which combines p l1-regularized node-wise regression problems together. The change point estimator and the corresponding estimated coefficients of precision matrices are obtained together. Based on that, a classifier is introduced to distinguish whether a change point exists. To recover the graphical structure correctly, a data-driven thresholding procedure is proposed. In theory, under some sparsity conditions and regularity assumptions, our method can correctly choose a homogeneous or heterogeneous model with high accuracy. Furthermore, in the latter case with a change point, we establish estimation consistency of the change point estimator, by allowing the number of nodes being much larger than the sample size. Moreover, it is shown that, in terms of structure recovery of Gaussian graphical models, the proposed thresholding procedure achieves model selection consistency and controls the number of false positives. The validity of our proposed method is justified via extensive numerical studies. Finally, we apply our proposed method to the S&P 500 dataset to show its empirical usefulness.
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Drawing inferences for High‐dimensional Linear Models: A Selection‐assisted Partial Regression and Smoothing Approach
Inference for high-dimensional models is challenging as regular asymptotic the- ories are not applicable. This paper proposes a new framework of simultaneous estimation and inference for high-dimensional linear models. By smoothing over par- tial regression estimates based on a given variable selection scheme, we reduce the problem to a low-dimensional least squares estimation. The procedure, termed as Selection-assisted Partial Regression and Smoothing (SPARES), utilizes data split- ting along with variable selection and partial regression. We show that the SPARES estimator is asymptotically unbiased and normal, and derive its variance via a non- parametric delta method. The utility of the procedure is evaluated under various simulation scenarios and via comparisons with the de-biased LASSO estimators, a major competitor. We apply the method to analyze two genomic datasets and obtain biologically meaningful results.
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- Award ID(s):
- 1712962
- NSF-PAR ID:
- 10106524
- Date Published:
- Journal Name:
- Biometrics
- ISSN:
- 1541-0420
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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We consider a semiparametric additive partially linear regression model (APLM) for analysing ultra‐high‐dimensional data where both the number of linear components and the number of non‐linear components can be much larger than the sample size. We propose a two‐step approach for estimation, selection, and simultaneous inference of the components in the APLM. In the first step, the non‐linear additive components are approximated using polynomial spline basis functions, and a doubly penalized procedure is proposed to select nonzero linear and non‐linear components based on adaptive lasso. In the second step, local linear smoothing is then applied to the data with the selected variables to obtain the asymptotic distribution of the estimators of the nonparametric functions of interest. The proposed method selects the correct model with probability approaching one under regularity conditions. The estimators of both the linear part and the non‐linear part are consistent and asymptotically normal, which enables us to construct confidence intervals and make inferences about the regression coefficients and the component functions. The performance of the method is evaluated by simulation studies. The proposed method is also applied to a dataset on the shoot apical meristem of maize genotypes.
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