# Tail dependence and smoothness

## TL;DR

This paper introduces a smoothness coefficient to assess oscillations in stochastic processes, linking it to tail dependence, and proposes a new estimator for tail dependence with evaluation through simulations.

## Contribution

It proposes a novel smoothness coefficient for processes, providing an intuitive measure of oscillations, and establishes its connection to tail dependence, leading to a new estimator.

## Key findings

- The smoothness coefficient coincides with tail dependence in stationary processes.
- A new estimator for tail dependence is developed and tested.
- Simulation results demonstrate the estimator's performance.

## Abstract

The risk of catastrophes is related to the possibility of occurring extreme values. Several statistical methodologies have been developed in order to evaluate the propensity of a process for the occurrence of high values and the permanence of these in time. The extremal index $\theta$ (Leadbetter 1983) allows to infer the tendency for clustering of high values, but does not allow to evaluate the greater or less amount of oscillations in a cluster. The estimation of $\theta$ entails the validation of local dependence conditions regulating the distance between high levels oscillations of the process, which is difficult to implement in practice. In this work, we propose a smoothness coefficient to evaluate the degree of smoothness/oscillation in the trajectory of a process, with an intuitive reading and simple estimation. Application in some examples will be provided. We will see that, in a stationary process, it coincides with the tail dependence coefficient $\lambda$ (Sibuya 1960, Joe 1997), providing a new interpretation of the latter. This relationship will inspire a new estimator for $\lambda$ and its performance will be evaluated based on a simulation study.