Statistics and Its Interface

Volume 4 (2011)

Number 2

An extension of max autoregressive models

Pages: 253 – 266



Philippe Naveau (Laboratoire des Sciences du Climat et de l’Environnement, LSCE-IPSL-CNRS, Gif-sur-Yvette, France)

Zhengjun Zhang (Department of Statistics, University of Wisconsin, Madison, Wi., U.S.A.)

Bin Zhu (Department of Statistics, University of Wisconsin, Madison, Wi., U.S.A.)


To model clustered maxima behaviors in time series analysis, max-autoregressive (MAR) and moving maxima (MM) processes are naturally adapted from linear autoregressive (AR) and moving average (MA) models. Yet, applications of MAR and MM processes are still sparse due to some difficulties of parameter inference and some abnormality of the processes. Basically, some ratios of observations can take constant values in MAR models. The objective of this present work is to introduce a new model that is closely related to the MAR processes and is free of the aforementioned abnormality. A logarithm transformation of the new model leads to time series models with log-positive alpha stable noises and hidden max Gumbel shocks. Theoretical properties of the new models are derived.


extreme value theory, dependence, Gumbel distribution, autoregressive model

2010 Mathematics Subject Classification

Primary 60G70, 62M10. Secondary 62G32.

Published 22 June 2011