Pareto processes

Abstract

An autoregressive process ARP(1) with Pareto-distributed inputs, analogous to those of Lawrance and Lewis (1977), (1980), is defined and its properties developed. It is shown that the stationary distributions are Pareto. Further, the maximum and minimum processes are asymptotically Weibull, and the ARP(1) process is shown to be closed under maximization or minimization when the number of terms is geometrically distributed. The ARP(1) process leads naturally to an extremal process in the sense of Lamperti (1964). Statistical inference for the ARP(1) process is developed. An absolutely continuous variant of the Pareto process is described.

A Gini Autocovariance Function for Time Series Modelling

“…That is, we give explicit closed‐form expressions for γ ( G ) (0), γ ( G ) (−1) and γ ( G ) (+1) in terms of the model parameters σ , p and α , as follows. First, as shown by Yeh et al (), we have Since F is continuous, we have E F ( X 1 ) = 1/2, and hence, our goal becomes the evaluation of reducing the problem to the evaluation of E ( X 1 + k F ( X 1 )). Deferring details of proof to the Appendix, here we discuss the solution and its interpretations relative to the parameters α and p of YARP(III)(1).…”

“…Here, we examine the Gini ACV for a nonlinear type of AR process with possibly infinite variance. Yeh et al () introduce, in different notation, a nonlinear AR Pareto process YARP(III)(1) given by where 0 < p < 1, with { ε t } i.i.d. from the Pareto distribution having survival function with the parameters μ , σ > 0 and α > 0 corresponding to location, scale and tail index respectively.…”

“…Under second‐order assumptions ( α > 2), the usual ACV is defined but problematic. Yeh et al () give an explicit formula for γ (0) but only an implicit formula involving an incomplete beta integral for γ (1) and do not treat γ ( k ), k ≥2. Therefore, alternative approaches have been developed to explore quantitatively the dependence features of YARP(III)(1).…”

“…Section 4 applies the Gini ACV to a nonlinear AR time series model of Pareto type introduced by Yeh et al () and treated recently by Ferreira (). For this model, no closed‐form expressions exist for the usual ACF when it is defined under second‐moment assumptions.…”

Gini autocovariance function used for time series with heavy‐tail distributions

“…Another application of the Gini method is for nonlinear type of AR process with possibly infinite variance. Such a model was introduced by Yeh, Arnold, and Robertson (). Their nonlinear AR Pareto process YARP(III)(1) is given by where 0 < p < 1, with { ε t } i.i.d.…”