Debris flows are rare and complex natural hazards, posing substantial challenges for statistical modelling due to data scarcity and the intricate interplay between rainfall, soil saturation, and geomorphological characteristics. Leveraging a recently developed database of debris-flow events in South Tyrol, this work explores statistical approaches for analysing such phenomena. We first discuss the limitations of standard regression-based methods for rare events, highlighting the challenges arising in highly sparse settings. Particular attention is then given to temporal dependence in environmental drivers. To this end, we propose a copula-based framework to model lagged dependence between debris-flow magnitude and antecedent rainfall. In this approach, the dependence structure is parameterized as a function of the lag, allowing us to capture delayed rainfall effects and to identify the temporal window most strongly associated with observed volumes of displaced sediments.