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Basic properties and prediction of max-ARMA processes

Published online by Cambridge University Press:  01 July 2016

Richard A. Davis  and
Sidney I. Resnick
Richard A. Davis*
Affiliation:
Colorado State University
Sidney I. Resnick*
Affiliation:
Cornell University
*
∗Postal address: Department of Statistics, Colorado State University, Fort Collins, CO 80523, USA.
∗∗Postal address: Department of Operations Research and Industrial Engineering, Upson Hall, Cornell University, Ithaca, NY 14853, USA.
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Abstract

A max-autoregressive moving average (MARMA(p, q)) process {Xt} satisfies the recursion for all t where φ i, , and {Zt} is i.i.d. with common distribution function Φ1,σ (X): = exp {–σ x–1} for . Such processes have finite-dimensional distributions which are max-stable and hence are examples of max-stable processes. We provide necessary and sufficient conditions for existence of a stationary solution to the MARMA recursion and we examine the reducibility of the process to a MARMA(p′, q′) with p′ <p or q′ < q. After introducing a natural metric between two jointly max-stable random variables, we consider the prediction problem for MARMA processes. Assuming that X1, …, Xn have been observed, we restrict our class of predictors to be max-linear, i.e. of the form , and find b1, …, bn to minimize the distance between this predictor and Xn+k for k 1. The optimality criterion is designed to minimize the probability of large errors and is similar in spirit to the dispersion criterion adopted in Cline and Brockwell (Stoch. Proc. Appl. 19 (1985), 281-296) for the prediction of ARMA processes with stable noise. Most of our results remain valid for the case when the distribution of Z1 is only in the domain of attraction of Φ1,σ. In addition, we give a naive estimation procedure for the φ 's and the θ 's which, with probability 1, identifies the true parameter values exactly for n sufficiently large.

Keywords

MAX-STABLE DISTRIBUTIONMAX-STABLE PROCESSESPREDICTIONCAUSALITYDOMAIN OF ATTRACTION OF AN EXTREME VALUE DISTRIBUTIONARMA PROCESSES
Type
Research Article
Information
Advances in Applied Probability , Volume 21 , Issue 4 , December 1989 , pp. 781 - 803
Copyright
Copyright © Applied Probability Trust 1989 

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Footnotes

Research supported by NSF Grant DMS-8802559, and partly carried out at the University of California, San Diego.

Research supported by NSF Grant DMS-8801034 and the Mathematical Sciences Institute at Cornell University.

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