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.
Highlights
Increased exposure to catastrophic losses and the complexity of financial instruments require sophisticated risk assessment tools in areas such as insurance, banking, finance, among others.Extreme value theory plays an important methodological role in risk management by providing appropriate instruments to deal with values as large as or even higher than those ever observed.These techniques include heavy-tailed models and measures to evaluate tail dependence, namely to infer to what extent the occurrence of a risk value in some variable influences an analogous occurrence in another variable.Linear ARMA with heavy-tailed noise may be suitable to model time series presenting peaks of observations
Max-autoregressive and moving maximum models were developed within this spirit, such as MARMA introduced in Davis and Resnick (1989) and M4 processes presented in Smith and Weissman (1996)
The Pareto model, which is closed under geometric multiplication and minimization, motivated the first-order Pareto processes presented in Arnold (2001)
Summary
Increased exposure to catastrophic losses and the complexity of financial instruments require sophisticated risk assessment tools in areas such as (re) insurance, banking, finance, among others. This is a model within the heavy-tailed class where mean values (of different orders) may not exist. In heavy-tailed models, the extremal observations are important, and a dependence analysis based on central measures like the most common autocorrelation may be misleading if the dependence in the tails presents a different structure from the remaining. The rate of the convergence, usually denoted coefficient of asymptotic tail independence η (Ledford and Tawn (1996); Wadsworth and Tawn (2012)) captures a residual tail dependence, revealing a kind of “penultimate” clustering, i.e., an aggregation of not so high values This is a not so fortuitous behavior in real applications and can be observed in the well-known Gaussian.
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