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December 2018 ROCKET: Robust confidence intervals via Kendall’s tau for transelliptical graphical models
Rina Foygel Barber, Mladen Kolar
Ann. Statist. 46(6B): 3422-3450 (December 2018). DOI: 10.1214/17-AOS1663


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.


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Rina Foygel Barber. Mladen Kolar. "ROCKET: Robust confidence intervals via Kendall’s tau for transelliptical graphical models." Ann. Statist. 46 (6B) 3422 - 3450, December 2018.


Received: 1 February 2016; Revised: 1 April 2017; Published: December 2018
First available in Project Euclid: 11 September 2018

MathSciNet: MR3852657
Digital Object Identifier: 10.1214/17-AOS1663

Primary: 62G10
Secondary: 62F12 , 62G20

Keywords: covariance selection , graphical model selection , post-model selection inference , rank-based estimation , transelliptical graphical models , uniformly valid inference

Rights: Copyright © 2018 Institute of Mathematical Statistics


Vol.46 • No. 6B • December 2018
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