Estimating Long Tail Models for Risk Trends

This letter develops a method for estimating trends of extreme events statistics across multiple time periods. Some of the periods might have no extreme events and some might have much data. The extreme event distribution is modeled with a Pareto or exponential tail. The method requires selecting an extreme event threshold and then solving two convex problems for the tail parameters. Solving one provides a smoothed tail rate trend, solving another, the smoothed trend of the tail quantile level. The approach is illustrated by trending the 10-year extreme event risks for S&P 500 index daily losses and for peak power load in electrical utility data.