On Robust Asymptotic Theory of Unstable AR(p) Processes with Infinite Variance

Abstract

In this thesis, we explore some asymptotic results in heavy-tailed theory. There are many empirical and compelling evidence in statistics that require modeling with heavy tailed observations. This thesis is divided into three parts. First, we consider a robust estimation of the mean vector for a sequence of independent and identically distributed observations in the domain of attraction of a stable law with possibly different indices of stability between 1 and 2. The suggested estimator is asymptotically normal with unknown parameters. We apply an asymptotically valid bootstrap to construct a confidence region for the mean vector. Furthermore, a simulation study is performed to show that the estimation method is efficient for conducting inference about the mean vector for multivariate heavy-tailed observations. In the second part, we present the asymptotic distribution of M-estimators for parameters in an unstable AR(p) process. The innovations are assumed to be in the domain of attraction of a stable law with index 0 < α ≤ 2. In particular, when the model involves repeated unit roots or conjugate complex unit roots, M- estimators have a higher asymptotic rate of convergence compared to the least square estimators. Moreover, we show that the asymptotic results can be written as Ito stochastic integrals. Finally, the preceding methodologies lead to develop the asymptotic theory of M-estimators for parameters in unstable AR(p) processes with nonzero location parameter. Similar to the preceding cases, we assume that the process is driven by innovations in the domain of attraction of a stable law with index 0 < α ≤ 2. In this thesis, for all models, we also cover the finite variance case (α = 2).