Extreme Value Analysis based on Blockwise Top-Two Order Statistics

TL;DR

This paper develops a top-two order statistics framework for extreme value analysis of time series by leveraging blockwise two largest observations per block. It shows that naive independence-based MLEs are generally inconsistent and introduces bias-corrected TT estimators that are consistent and often more efficient than block-max approaches, especially for estimating tail-shape parameters and return levels. The authors derive both disjoint and sliding block theory, provide iid and dependent data results, and demonstrate improved performance through extensive simulations and a climate-case study, with circular-block bootstrap enabling practical uncertainty quantification. The work suggests promising extensions to top-m order statistics and to more flexible parametric families, offering a refined toolkit for environmental and other time-series extremes.

Abstract

Extreme value analysis for time series is often based on the block maxima method, in particular for environmental applications. In the classical univariate case, the latter is based on fitting an extreme-value distribution to the sample of (annual) block maxima. Mathematically, the target parameters of the extreme-value distribution also show up in limit results for other high order statistics, which suggests estimation based on blockwise large order statistics. It is shown that a naive approach based on maximizing an independence log-likelihood yields an estimator that is inconsistent in general. A consistent, bias-corrected estimator is proposed, and is analyzed theoretically and in finite-sample simulation studies. The new estimator is shown to be more efficient than traditional counterparts, for instance for estimating large return levels or return periods.

Figures (26)

• Figure 1: Different $\rho$ functions. The examples 'linear', 'power' and 'ARMAX' correspond to Example \ref{['ex:models']} [a] ($c=0.6$), [b] ($c=0.4$) and [c] ($c=0.6$), respectively.
• Figure 2: Left: graph of $\rho_0\mapsto \varpi_{\rho_0}$. Right: graph of its derivative.
• Figure 3: Standardized asymptotic variance of shape (left) and scale (right) estimators, that is, the diagonal entries of the asymptotic covariance matrices $\Sigma^{({ {\operatorname{mb} }})}_{{{ {\operatorname{TopTwo}}}}}(1, \rho)$ and $\Sigma^{({ {\operatorname{mb} }})}_\mathrm{max}(1)$ from \ref{['eq:sigma-disjoint-general']} and \ref{['eq:max_dbm_sbm_matrix']}, respectively, at $\alpha_0=1$ and as a function of $\rho_0$. The examples "linear", "power" and "kink" correspond to Example \ref{['ex:models']} [a], [b] and [c], respectively. For the disjoint blocks version, the respective curves for an arbitrary $\rho \in \mathcal{C}$ lie between the 'linear" and "kink" curves.
• Figure 4: Standardized asymptotic bias of shape (left) and scale (right) estimators as a function of $\rho_0$, for $\alpha_0=1$ and $\lambda_1=1$. More precisely, the depicted values correspond to the mean of the asymptotic distributions of $\sqrt k_n(\widetilde{\alpha}_n - \alpha_0)$ and $\sqrt k_n(\widetilde{\sigma}_n/\sigma_n - 1)$, respectively, under the assumption that $\sqrt{k_n}/r_n = \lambda_1+o(1)$ for $n \to \infty$.
• Figure 5: Asymptotic MSE of $\widetilde{\alpha}_n^{({ {\operatorname{mb} }})}$ as a function of the block size $r$, for fixed $\alpha_0=1$, $n = 1000$ and three choices of $\rho(\eta)=c\cdot(1-\eta)$, $c\in\{0.2, 0.5, 0.9\}$.

Theorems & Definitions (67)

• Theorem 2.1: Welsch_1972
• Remark 2.2: The Fréchet-Welsch-distribution
• Example 2.3: Stationary time series and models for $\rho$
• Lemma 3.1: Existence and uniqueness
• Lemma 3.2
• Theorem 3.4: (Lack of) consistency
• Remark 3.5: An alternative pseudo-maximum likelihood estimator
• Theorem 3.7: Asymptotic Distribution
• Theorem 3.8: Consistency of the bias-corrected estimator
• proof