ROCKET: Robust Confidence Intervals via Kendall's Tau for Transelliptical Graphical Models

TL;DR

This paper introduces ROCKET, a new method for constructing confidence intervals in transelliptical graphical models, effectively capturing heavy tail dependence and outperforming existing Gaussian-based models in both simulated and real data.

Contribution

The paper proposes the ROCKET method, providing asymptotically normal estimators for transelliptical models, extending graphical model inference beyond Gaussian assumptions.

Findings

ROCKET achieves asymptotic normality under mild conditions.

ROCKET outperforms nonparanormal and Gaussian models in simulations.

On real stock data, ROCKET's behavior aligns with theoretical predictions.

Abstract

Undirected graphical models are used extensively in the biological and social sciences to encode a pattern of conditional independences between variables, where the absence of an edge between two nodes $a$ and $b$ indicates that the corresponding two variables $X_a$ and $X_b$ are believed to be conditionally independent, after controlling for all other measured variables. In the Gaussian case, conditional independence corresponds to a zero entry in the precision matrix $\Omega$ (the inverse of the covariance matrix $\Sigma$). Real data often exhibits heavy tail dependence between variables, which cannot be captured by the commonly-used Gaussian or nonparanormal (Gaussian copula) graphical models. In this paper, we study the transelliptical model, an elliptical copula model that generalizes Gaussian and nonparanormal models to a broader family of distributions. We propose the ROCKET method, which constructs an estimator of $\Omega_{ab}$ that we prove to be asymptotically normal under mild assumptions. 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.