[Paper Review] Generation and Detection of Multivariate Regular Variation and Hidden Regular Variation

Bikramjit Das, Sidney I. Resnick|arXiv (Cornell University)|Mar 23, 2014

Financial Risk and Volatility Modeling19 references3 citations

TL;DR

This paper proposes methods for generating and detecting multivariate regular variation (MRV) and hidden regular variation (HRV) in heavy-tailed data, using mixture, multiplication, and additive generation techniques. It demonstrates that HRV can be detected via conditional extreme value (CEV) models after generalized polar transformation, with empirical validation on Internet traffic data showing consistent HRV structure and tail index estimates between 1 and 2.

ABSTRACT

We review definitions of multivariate regular variation (MRV) and hidden regular variation (HRV) for distributions of random vectors and then summarize methods for generating models exhibiting both properties. We also discuss diagnostic techniques that detect these properties in multivariate data and indicate when models exhibiting both MRV and HRV are plausible fits for the data. We illustrate our techniques on simulated data and also two real Internet data sets.

Motivation & Objective

• To develop practical, implementable methods for generating multivariate regularly varying distributions with both MRV and HRV properties.
• To provide diagnostic tools for detecting MRV and HRV in real multivariate data, especially when marginal tails are heavy but asymptotic dependence is subtle.
• To address identifiability issues in additive generation methods where asymptotic parameters may not correspond to intended components.
• To validate detection techniques on real Internet traffic data, demonstrating HRV presence through angular and tail index diagnostics.
• To extend the applicability of CEV models to detect HRV on cones like ℝ₊²∖{axes}, enabling structured tail dependence modeling.

Proposed method

• Uses three generation methods—mixture, multiplication, and additive—each designed to produce joint distributions with specified MRV and HRV asymptotic limits on ℝ₊²∖{0} and ℝ₊²∖{axes}.
• Applies generalized polar coordinate transformation to convert multivariate regular variation into a conditional extreme value (CEV) model, enabling use of existing CEV diagnostics.
• Employs Hill plots and Pickandsish/Hillish diagnostics on transformed variables to test consistency with CEV models and estimate tail indices.
• Uses rank-transformed data and order statistics to estimate the proportion q of HRV contribution and to assess angular measures G₁ and G₂.
• Applies QQ plots of log-ratios (e.g., log(S*/R*)) to detect heavy-tailed behavior in the hidden angular component.
• Validates detection via kernel density estimation and histograms of ratios under high-threshold conditions (e.g., Aᵢ > A₍₁₀₀₎).

Experimental results

Research questions

• RQ1 Can we generate multivariate models that exhibit both multivariate regular variation (MRV) and hidden regular variation (HRV) with controlled asymptotic parameters?
• RQ2 How can we detect HRV in real multivariate data when standard marginal tail checks are insufficient?
• RQ3 What diagnostic tools are effective in identifying HRV structure, especially when the hidden dependence is not visible in marginal distributions?
• RQ4 To what extent do additive generation methods preserve parameter identifiability, and what are the risks of misestimation?
• RQ5 Can the CEV model framework be reliably applied to detect HRV in real-world Internet traffic data?

Key findings

• The additive generation method, while effective in theory, suffers from identifiability issues where asymptotic parameters may not reflect the true component distributions.
• Empirical analysis of UNC Internet data shows tail indices for S and R between 1 and 2, indicating heavy-tailed marginals consistent with regular variation.
• Angular density plots indicate asymptotic independence, but HRV is detected via Hill plots of min(S*, R*) showing tail index estimates greater than 1.
• Hillish and Pickandsish diagnostics on transformed variables (A, θ₁) and (A, θ₂) show stable behavior around 1 and 0, respectively, supporting CEV model consistency.
• Estimation of the hidden angular measure reveals q̂ ≈ 0.55, indicating roughly equal contribution from both components in the HRV regime.
• QQ plots and Hill plots of log(S*/R*) suggest light-tailed behavior, while those of log(R*/S*) indicate heavy-tailed distributions with tail index between 1 and 1.5, supporting HRV structure.