On the extremal behavior of a Pareto process: an alternative for ARMAX modeling (Q2893932)

Any stochastic process whose marginal distributions are of the Pareto or generalized Pareto form is called a Pareto process. This paper considers the extremal behavior of the Yeh-Arnold-Robertson Pareto (III) process, where the marginal distributions are of the form NEWLINE\[NEWLINE F\left( x\right) =1-\left[ 1+\left( \frac{x-\mu}{\sigma}\right) ^{\alpha }\right] ^{-1},\quad x>\mu, NEWLINE\]NEWLINE where \(\sigma>0\) is a scale parameter, \(\mu\) is a location parameter and \(\alpha>0\) is a shape parameter. The author presents a complete characterization of the tail behavior of the Yeh-Arnold-Robertson Pareto (III) process. She concludes that it is similar to the first order max-autoregressive ARMAX processes, but has a more robust parameter estimation procedure. Therefore, the Yeh-Arnold-Robertson Pareto (III) process is an attractive alternative to ARMAX modeling if we are interested in the tails. Consistency and asymptotic normality ot the presented estimators are proved.