ROCKET: robust confidence intervals via Kendall’s tau for transelliptical graphical models. Understanding complex relationships between random variables is of fundamental importance in high-dimensional statistics, with numerous applications in biological and social sciences. Undirected graphical models are often used to represent dependencies between random variables, where an edge between two random variables is drawn if they are conditionally dependent given all the other measured variables. A large body of literature exists on methods that estimate the structure of an undirected graphical model, however, little is known about the distributional properties of the estimators beyond the Gaussian setting. In this paper, we focus on inference for edge parameters in a high-dimensional transelliptical model, which generalizes Gaussian and nonparanormal graphical models. We propose ROCKET, a novel procedure for estimating parameters in the latent inverse covariance matrix. We establish asymptotic normality of ROCKET in an ultra high-dimensional setting under mild assumptions, without relying on oracle model selection results. ROCKET requires the same number of samples that are known to be necessary for obtaining a (sqrt{n}) consistent estimator of an element in the precision matrix under a Gaussian model. Hence, it is an optimal estimator under a much larger family of distributions. The result hinges on a tight control of the sparse spectral norm of the nonparametric Kendall’s tau estimator of the correlation matrix, which is of independent interest. Empirically, ROCKET outperforms the nonparanormal and Gaussian models in terms of achieving accurate inference on simulated data. We also compare the three methods on real data (daily stock returns), and find that the ROCKET estimator is the only method whose behavior across subsamples agrees with the distribution predicted by the theory.

References in zbMATH (referenced in 13 articles , 1 standard article )

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  1. Dai, Ran; Kolar, Mladen: Inference for high-dimensional varying-coefficient quantile regression (2021)
  2. Rossell, David; Zwiernik, Piotr: Dependence in elliptical partial correlation graphs (2021)
  3. Eisenach, Carson; Bunea, Florentina; Ning, Yang; Dinicu, Claudiu: High-dimensional inference for cluster-based graphical models (2020)
  4. Liu, Yu; Ren, Zhao: Minimax estimation of large precision matrices with bandable Cholesky factor (2020)
  5. Niu, Lu; Liu, Xiumin; Zhao, Junlong: Robust estimator of the correlation matrix with sparse Kronecker structure for a high-dimensional matrix-variate (2020)
  6. Yu, Ming; Gupta, Varun; Kolar, Mladen: Simultaneous inference for pairwise graphical models with generalized score matching (2020)
  7. He, Yong; Zhang, Liang; Ji, Jiadong; Zhang, Xinsheng: Robust feature screening for elliptical copula regression model (2019)
  8. Li, Ang; Barber, Rina Foygel: Multiple testing with the structure-adaptive Benjamini-Hochberg algorithm (2019)
  9. Barber, Rina Foygel; Kolar, Mladen: ROCKET: robust confidence intervals via Kendall’s tau for transelliptical graphical models (2018)
  10. Lu, Junwei; Kolar, Mladen; Liu, Han: Post-regularization inference for time-varying nonparanormal graphical models (2018)
  11. Neykov, Matey; Ning, Yang; Liu, Jun S.; Liu, Han: A unified theory of confidence regions and testing for high-dimensional estimating equations (2018)
  12. Hirose, Kei; Fujisawa, Hironori; Sese, Jun: Robust sparse Gaussian graphical modeling (2017)
  13. Cai, T. Tony; Ren, Zhao; Zhou, Harrison H.: Rejoinder of “Estimating structured high-dimensional covariance and precision matrices: optimal rates and adaptive estimation” (2016)