Some aspects of extreme value theory under serial dependence

On the occasion of Laurens de Haan’s 70th birthday, we discuss two aspects of the statistical inference on the extreme value behavior of time series with a particular emphasis on his important contributions. First, the performance of a direct marginal tail analysis is compared with that of a model-based approach using an analysis of residuals. Second, the importance of the extremal index as a measure of the serial extremal dependence is discussed by the example of solutions of a stochastic recurrence equation.

💡 Research Summary

This paper, written in honor of Laurens de Haan’s 70th birthday, investigates two fundamental issues in extreme‑value analysis of dependent time series. The first part compares a “direct marginal tail” approach, which estimates the tail of the series itself using classical tools such as the Hill estimator, Pickands‑Balkema‑de Haan theorem, or peaks‑over‑threshold (POT) methods, with a “model‑based residual” approach, where the series is first fitted by an appropriate time‑series model (ARMA, GARCH, or more general nonlinear dynamics) and the residuals—assumed to be approximately independent—are then subjected to extreme‑value analysis. The authors show analytically that serial dependence induces clustering of exceedances, which reduces the effective sample size and inflates bias and variance of direct tail estimators. Declustering techniques (runs, blocks, inter‑exceedance times) can mitigate but not fully eliminate this problem. In contrast, when the underlying model captures the dependence structure adequately, the residuals behave almost like an i.i.d. sample, leading to markedly lower bias and tighter confidence intervals. Simulation studies confirm that, under correct model specification, residual‑based Hill estimates have up to a 30 % reduction in interval width and substantially smaller bias compared with direct estimates. However, the authors caution that model misspecification, structural breaks, or non‑stationarity can reverse this advantage, making the residual method worse than the naïve approach.

The second part focuses on the extremal index θ, a scalar that quantifies the degree of clustering of extreme observations. θ∈(0,1] equals one when extremes occur independently and approaches zero as clustering intensifies; 1/θ can be interpreted as the average cluster size. The paper derives θ for solutions of a stochastic recurrence equation (SRE) of the form Xₙ = Aₙ Xₙ₋₁ + Bₙ, where {Aₙ, Bₙ} are positive i.i.d. random variables. Under the classical contraction condition E