Journal of Data Science

Satellite precipitation products have the potential to be employed for the purpose of better understanding extreme precipitation events in remote mountainous terrain, where weather stations and radar data tend to be sparse. For this reason, it is crucial to assess how closely satellite estimates agree with ground observations during extreme events, and how that agreement varies across such regions. We use asymptotic dependence from multivariate extreme value theory as the primary tool in this study. After presenting two measures of asymptotic dependence and their associated estimators, we illustrate these ideas using simulated data. We then model the level of asymptotic dependence between PERSIANN-CDR and SNOTEL station data over the US Northern Rocky Mountains. We consider both asymptotic dependence estimators, and based on hypothesis tests and visual diagnostics, both estimates of asymptotic dependence indicate positive spatial dependence. We also investigate whether geographical factors influence the levels of asymptotic dependence over this region. Using a spatial correlation analysis, we find that elevation is negatively correlated with both asymptotic dependence estimators and average summer temperature is positively correlated with both asymptotic dependence estimators. However, we did not find any geographical covariates to be statistically significant in the model.

Our contribution is to widen the scope of extreme value analysis applied to discrete-valued data. Extreme values of a random variable are commonly modeled using the generalized Pareto distribution, a peak-over-threshold method that often gives good results in practice. When data is discrete, we propose two other methods using a discrete generalized Pareto and a generalized Zipf distribution respectively. Both are theoretically motivated and we show that they perform well in estimating rare events in several simulated and real data cases such as word frequency, tornado outbreaks and multiple births.

In this work, we introduce a new distribution for modeling the extreme values. Some important mathematical properties of the new model are derived. We assess the performance of the maximum likelihood method in terms of biases and mean squared errors by means of a simulation study. The new model is better than some other important competitive models in modeling the repair times data and the breaking stress data.