Analyzing the Gaver—Lewis Pareto Process under an Extremal Perspective

Abstract

Abstract:Pareto processes are suitable to model stationary heavy-tailed data. Here, we consider the auto-regressive Gaver-Lewis Pareto Process and address a study of the tail behavior. We characterize its local and long-range dependence. We will see that consecutive observations are asymptotically tail independent, a feature that is often misevaluated by the most common extremal models and with strong relevance to the tail inference. This also reveals clustering at "penultimate" levels. Linear correlation may not exist in a heavy-tailed context and an alternative diagnostic tool will be presented. The derived properties relate to the auto-regressive parameter of the process and will provide estimators. A comparison of the proposals is conducted through simulation and an application to a real dataset illustrates the procedure.

Active Management of Operational Risk in the Regimes of the “Unknown”: What Can Machine Learning or Heuristics Deliver?

“…As already pointed out by Moscadelli (2004); De Fontnouvelle et al (2005); Giacometti et al (2007), such data for severity S (equaling monetary loss due to OpRisk events) and frequency F(S) can be fitted with a power law distribution F(S) ∝ S −λ . For a more detailed analysis, a Generalized Pareto Distribution (GPD; see, e.g., Ferreira and Haan 2014;Ferreira and Ferreira 2017)) or other sophisticated statistical distributions can be applied, but for the scope of this paper, a power law provides the simplest fit to data with "fat tails" following Occam's razor and reveals the fundamental difference in the shape of the distribution to the "peaked" Gaussian-like distribution used for market and credit risk.…”