Subsampling Weakly Dependent Times Series and Application to Extremes

• Activity Number: 309
• Type: Contributed
• Date/Time: Tuesday, August 2, 2011 : 8:30 AM to 10:20 AM
• Sponsor: IMS
• Author(s): Silika Prohl*+ and Paul Doukhan and Christian P. Robert
• Companies: University of Zurich/Princeton University and University Cergy-Pontoise and Universite Paris-Dauphine
• Keywords: Extremes ; Subsampling ; Weak dependence ; Mixing

Abstract

Politis and Romano (1994) established the subsampling estimator for converging statistics when the underlying sequence is strongly mixing. Bertail \textit{et al.} (2004) applied this work to subsampling estimators for distributions of diverging statistics. In particular, they constructed an approximation of the distribution of the sample maximum without any information on the tail of the stationary distribution. However, the assumption on the strong mixing properties of the time series is sometimes too strong as for the class of first-order autoregressive sequences with uniform marginal distribution introduced and studied by Chernick (1981): let $(X_{t})_{t\in \mathbb{Z}}$ be uniform AR(1) process defined recursively as \begin{equation} X^{(r)}_{t}=\frac{1}{r} X^{(r)}_{t-1}+\varepsilon _{t}, \label{FOS} \end{equation} where $r\geq 2$ is an integer, $(\varepsilon _{t})_{t\in \mathbb{Z}}$ are iid and uniformly distributed on the set $\{0,1,\ldots ,r-1\}$ and $X^{(r)}_{0}$ is uniformly distributed on $[0,1]$. The results of Bertail \textit{et al.} (2004) can not be used for this class of processes although the normalized sample maximum has a non-degenerate limiting distribution