Jorge Castillo-Mateo, Alan E. Gelfand, Ana C. Cebrián, and Jesús Asín’s contribution to the Discussion of ‘Inference for extreme spatial temperature events in a changing climate with application to Ireland’ by Healy et al.

The authors have taken on the challenge of proposing a novel methodology to find new insights in extreme spatial temperature events. However, with a very strong word limitation, it is not possible to offer a proper discussion of their contribution. Instead, we briefly review a similar stream of work that our group has developed over the past five years. We also have been working with daily maximum temperature data, temporally over more than 60 years and spatially for both peninsular Spain and a subregion containing Aragón. We have developed our work primarily in the context of the incidence of extreme heat events, i.e. consecutive days above local space-time thresholds. We have focused on autoregressive spatial models with primary interest in assessing change in the incidence of extreme behaviour over time. Our initial effort (Schliep et al., 2021) focused on mean modelling, recognizing the need to model the bulk of the data as well as the upper tails of the data. With thresholds, the body of the data was specified through truncated normals and the upper tail was specified through t-distributions with autoregression in order to capture temporal tail dependence. With the spatio-temporal dependence structure, these models immediately inherited spatial tail dependence. Cebrián et al. (2022) introduced the idea of proportion of the space where temperature exceeded threshold. The authors pursue this spirit with their spatial risk measure in Section 5. Follow-on work appears in Castillo-Mateo et al. (2022) and Cebrián et al. (2023).