Dependence structures of Polya tree autoregressive models

Abstract

According to the recent developments of models for dependent data and autoregressive processes based on Polya trees (West, 1996), the dyadic expansion of an observation Xt, defined on a separable measurable space Omega, is used here to investigate the dependence structure of an autoregressive process of order one under non-normal marginal distributions of Xt. It is assumed that the dyadic representation of two consecutive observations is identical up to a level k (i.e. the last known matching point), with k distributed as a Geometric random variable with parameter Teta. Such parameter Teta models the autoregressive structure of the time series as a large k yields high dependence and a small k gives low dependence. In this work, we showed that small values of Teta produce an almost linear relation between xt and x(t+ 1), whereas Teta equal to one leads to independence, and values of Teta between zero and one allow for non-linear dependence. Moreover, we elicited the conditional distributions p(x(t+1)Ixt) and explore their features in correspondence of Uniform marginal distributions of Xt.