Statistics of Dependent Variables


Classical extreme value statistics is dominated by the theory for independent and identically distributed (iid) observations. In many applications, though, one encounters a non-negligible serial (or spatial) dependence. For instance, returns of an investment over successive periods are usually dependent, cf. Chapter 16, and stable low pressure systems can lead to extreme amounts of rainfall over several consecutive days. These examples demonstrate that a positive dependence between extreme events is often particularly troublesome as the consequences, which are already serious for each single event, may accumulate and finally result in a devastating catastrophe.


Dependence Structure Time Series Model Asymptotic Variance GARCH Model Extremal Index 
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  1. 2.
    Hsing, T. (1991). Estimating the parameters of rare events. Stoch. Processes Appl. 37, 117–139.MATHCrossRefMathSciNetGoogle Scholar
  2. 3.
    Gomes, M.I. and de Haan, L. (2003). Joint exceedances of the ARCH process. Preprint, Erasmus University Rotterdam.Google Scholar
  3. 4.
    Révész, P. (1968). The Laws of Large Numbers. Academic Press, New York.MATHGoogle Scholar
  4. 5.
    Pantula, S.G. (1988). Estimation of autoregressive models with ARCH errors. Sankhyā 50, 119–148.MATHMathSciNetGoogle Scholar
  5. 6.
    Drees, H. (2000). Weighted approximations of tail processes for β-mixing random variables. Ann. Appl. Probab. 10, 1274–1301.MATHMathSciNetGoogle Scholar
  6. 7.
    Drees, H. (2002). Tail empirical processes under mixing conditions. In: H.G. Dehling, T. Mikosch und M. Sørensen (eds.), Empirical Process Techniques for Dependent Data, 325–342, Birkhäuser, Boston.Google Scholar
  7. 8.
    Drees, H. (2003). Extreme quantile estimation for dependent data with applications to finance. Bernoulli 9, 617–657.MATHMathSciNetGoogle Scholar
  8. 9.
    Drees, H. (2003). Extreme quantile Estimation for Dependent Data with Applications to Finance. Bernoulli 9, 617–657.MATHMathSciNetGoogle Scholar
  9. 10.
    Drees, H. (2003). Extreme quantile stimation for dependent data with applications to finance. Bernoulli 9, p. 652.CrossRefMathSciNetGoogle Scholar
  10. 11.
    Datta, S. and McCormick, W.P. (1998). Inference for the tail parameters of a linear process with heavy tail innovations. Ann. Inst. Statist. Math. 50, 337–359.MATHCrossRefMathSciNetGoogle Scholar
  11. 12.
    Resnick, S.I. and Starica, C. (1997). Asymptotic behavior of Hill’s estimator for autoregressive data. Comm. Statist. Stochastic Models 13, 703–721.MATHCrossRefMathSciNetGoogle Scholar
  12. 13.
    Ling, S. and Peng, L. (2004). Hill’s estimator for the tail index of an ARMA model. J. Statist. Plann. Inference 123, 279–293.MATHCrossRefMathSciNetGoogle Scholar
  13. 14.
    Davis, R.A. and Resnick, S.I. (1986). Limit theory for the sample covariance and correlation functions of moving averages. Ann. Statist. 14, 533–558.MATHMathSciNetGoogle Scholar
  14. 15.
    Starica, C. and Pictet, O. (1997). The tales the tails of GARCH processes tell. Preprint, Chalmers University Gothenburg.Google Scholar
  15. 16.
    Mikosch, T. (2003). Modeling dependence and tails of financial time series. In: Extreme Value Theory and Applications, Finkenstadt, B. and Rootzén, H. (eds.), Chapman and Hall, 185–286.Google Scholar

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