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Estimation of $$\alpha$$-stable sub-Gaussian distributions for asset returns. (English) Zbl 1154.91601
Bol, Georg (ed.) et al., Risk assessment. Decisions in banking and finance. Selected papers based on the presentations at the 9th econometric workshop, Karlsruhe, Germany, April 5–7, 2006. Heidelberg : Physica-Verlag (ISBN 978-3-7908-2049-2/hbk; 978-3-7908-2050-8/ebook). Contributions to Economics, 111-152 (2009).
Summary: Fitting multivariate $$\alpha$$-stable distributions to data is still not feasible in higher dimensions since the (non-parametric) spectral measure of the characteristic function is extremely difficult to estimate in dimensions higher than 2. $$\alpha$$-stable sub-Gaussian distributions are a particular (parametric) subclass of the multivariate $$\alpha$$-stable distributions. We present and extend a method to estimate the dispersion matrix of an $$\alpha$$-stable sub-Gaussian distribution and estimate the tail index $$\alpha$$ of the distribution. In particular, we develop an estimator for the off-diagonal entries of the dispersion matrix that has statistical properties superior to the normal off-diagonal estimator based on the covariation. Furthermore, this approach allows estimation of the dispersion matrix of any normal variance mixture distribution up to a scale parameter. We demonstrate the behaviour of these estimators by fitting an $$\alpha$$-stable sub-Gaussian distribution to the DAX30 components. Finally, we conduct a stable principal component analysis and calculate the coefficient of tail dependence of the prinipal components.
For the entire collection see [Zbl 1155.91006].

##### MSC:
 91G70 Statistical methods; risk measures 62P05 Applications of statistics to actuarial sciences and financial mathematics 62E10 Characterization and structure theory of statistical distributions 62H25 Factor analysis and principal components; correspondence analysis
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