Conditional Extreme Value Estimation for Dependent Time Series

We study the consistency and weak convergence of the conditional tail function and conditional Hill estimators under broad dependence assumptions for a heavy-tailed response sequence and a covariate sequence. Consistency is established under $α$-mixing, while asymptotic normality follows from $β$-mixing and second-order conditions. A key aspect of our approach is its versatile functional formulation in terms of the conditional tail process. Simulations demonstrate its performance across dependence scenarios. We apply our method to extreme event modelling in the oil industry, revealing distinct tail behaviours under varying conditioning values.

💡 Research Summary

This paper develops a comprehensive theory for estimating conditional extreme‑value characteristics of a heavy‑tailed response time series when a covariate series is observed simultaneously. The authors assume that, for each covariate value x in an open set U, the conditional distribution of the response Y₀ given X₀ = x belongs to the regular variation class with tail index γ(x) > 0 and slowly varying function Lₓ(·). Under this conditional regular variation framework they introduce a non‑parametric Nadaraya–Watson kernel estimator for the conditional survival function Fₓ(y) and, based on a high‑threshold quantile qₙ,kₙ(x), define two key objects: (i) the conditional tail empirical function T̂ₓₙ(s) = F̂ₓₙ(s qₙ,kₙ(x))/Fₓ(uₙₓ) and (ii) a conditional Hill estimator γ̂ₙ(x) that mimics the classical Hill estimator but uses only observations whose covariate lies within a bandwidth hₙ of x. Because the effective sample size is n hₙ kₙ rather than kₙ, the authors carefully analyse how the bandwidth and the number of upper order statistics interact.

The first set of results establishes uniform consistency of T̂ₓₙ(s) on compact subsets of (0,∞). For m‑dependent sequences, consistency follows under mild regularity conditions (Theorem 1). For α‑mixing sequences with mixing coefficients α(j) = O(j^{−η}) and η > 2, consistency still holds provided an additional rate condition (3) linking the bandwidth, the tail probability, and the mixing decay (Theorem 2). These results rely on a γ‑Lipschitz continuity of 1/γ(·), a boundedness condition on the slowly varying functions, and a uniform bound on certain conditional expectations (Condition 1.6). As a corollary, the empirical quantile qₙ,kₙ(x) is shown to be asymptotically equivalent to the deterministic threshold uₙₓ, which in turn yields consistency of the conditional Hill estimator (Proposition 1, Theorem 3).

The second major contribution is a functional central limit theorem (CLT) for the centered tail process \