The Annals of Statistics

ROCKET: Robust confidence intervals via Kendall’s tau for transelliptical graphical models

Rina Foygel Barber and Mladen Kolar

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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.

Article information

Ann. Statist., Volume 46, Number 6B (2018), 3422-3450.

Received: February 2016
Revised: April 2017
First available in Project Euclid: 11 September 2018

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Mathematical Reviews number (MathSciNet)

Primary: 62G10: Hypothesis testing
Secondary: 62F12: Asymptotic properties of estimators 62G20: Asymptotic properties

Graphical model selection transelliptical graphical models covariance selection uniformly valid inference post-model selection inference rank-based estimation


Barber, Rina Foygel; Kolar, Mladen. ROCKET: Ro bust c onfidence intervals via Ke ndall’s t au for transelliptical graphical models. Ann. Statist. 46 (2018), no. 6B, 3422--3450. doi:10.1214/17-AOS1663.

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Supplemental materials

  • Supplement to “ROCKET: Robust emphconfidence intervals via Kendall’s tau for transelliptical graphical models”. In the supplementary materials, we provide additional experimental results (as described in Section 5), as well as details for all proofs of the theoretical results provided in this paper.