Estimation and inference for the persistence of extremely high temperatures

We propose a nonparametric framework for estimating the extremal index that captures the persistence of extreme observations. The framework provides unified and simple procedures for verifying the well-known local dependence condition $D^{(d)}(u_n)$, which characterizes the extremal index yet is often assessed through heuristic checks, and for selecting $d$ (a key parameter for estimation) when the condition holds. Under a general ω-mixing condition, we establish the asymptotic normality of the proposed estimator and prove the consistency of both the tuning parameter selection and the verification procedure for the $D^{(d)}(u_n)$ condition. Simulation studies show improved performance relative to two commonly used methods in terms of empirical mean squared errors. We analyze summer apparent temperature data for nine European cities from 1940 to 2025. The results show strong evidence of persistence in extreme temperatures for all cities, with such extremes typically lasting at least two days. The probability of two-day extreme-temperature events is two to four times higher in the most recent three decades relative to 1940–1974.