Probabilistic modeling of extreme events involving decaying variables with an application in seismology

Natural hazards are the extreme events that significantly distress life on Earth. To mitigate the detrimental impacts of these extreme events, it is essential to examine and model them using a probabilistic approach. Probability distributions are competent enough to analyze the exponential behavior and estimate the pattern of randomness in these real-life phenomena. We use the generalized Pareto-exponential distribution (GPED) and find it to be an appropriate model for extreme events that involve exponentially decaying variables. Interestingly, the GPED also comprises the features of both the well-known exponential and Pareto distributions and approaches several other well-known distributions after certain transformations. We derive its various probabilistic characteristics and provide an empirical study for different parametric values to observe their behavior. We follow the maximum likelihood method to estimate the unknown model parameters and conduct a simulation study for different sample sizes and different combinations of the model parameters to examine their stability. We also demonstrate the applicability of our model by using a data set from the field of seismology and establish its better performance by comparing it with some extant distributions.