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Title: Testing for a change in the tail parameter of regularly varying time series with long memory Authors:  Davide Giraudo - Ruhr-Universität Bochum (Germany) [presenting]
Annika Betken - Ruhr-Universitat Bochum (Germany)
Rafal Kulik - University of Ottawa (Canada)
Abstract: Let $(X_j)_{j\geq 1}$ be a strictly stationary sequence such that the distribution function $F$ of $X_1$ is regularly varying with parameter $-\alpha$, $\alpha>0$, i.e. $P(X>x)=x^{-\alpha}L(x)$, where $L$ slowly varying. One can show that $\lim_{u\rightarrow \infty} E\left(\log(X/u) X>u\right)=\lim_{u\rightarrow \infty} E\left(\log(X/u)1_{\{X>u\}}\right)/P(X>u)=1/\alpha=:\gamma$. Consequently, the parameter $\alpha$ can be estimated by $\hat\gamma=(1/\sum_{j=1}^n1_{\{X_j>u_n\}})\sum_{j=1}^n\log(X_j/u_n)1_{\{X_j>u_n\}}$, where $\left(u_n\right)_{n\geq 1}$ is a sequence such that $u_n\to\infty$ and $n\left(1-F(u_n)\right)\to\infty$. To this aim, a two-dimensional empirical process can be used. We will focus on the special case of a volatility model.