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On Functional Central Limit Theorems for Dependent, Heterogeneous Tail Arrays with Applications to Tail Index and Tail Dependence Estimators

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  • Jonathan Hill

    ()
    (Department of Economics, Florida International University)

Abstract

We establish functional central limit theorems for a broad class of dependent, heterogeneous tail arrays encountered in the extreme value literature, including extremal exceedances, tail empirical processes and tail empirical quantile processes. We trim dependence assumptions down to a minimum by constructing extremal versions of mixing and Near-Epoch-Dependence properties, covering mixing, ARFIMA, FIGARCH, bilinear, random coefficient autoregressive, nonlinear distributed lag and Extremal Threshold processes, and stochastic recurrence equations. Of practical importance our theory can be used to characterize the functional limit distributions of sample means and covariances of tail arrays, including popular tail index estimators, the tail quantile function, and multivariate extremal dependence measures under substantially general conditions.

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File URL: http://casgroup.fiu.edu/pages/docs/2244/1275230073_06-07.pdf
File Function: Revised version, 2006
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Bibliographic Info

Paper provided by Florida International University, Department of Economics in its series Working Papers with number 0607.

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Length: 31 pages
Date of creation: Jul 2006
Date of revision:
Handle: RePEc:fiu:wpaper:0607

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Postal: Miami, FL 33199
Phone: (305) 348-2316
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Web page: http://casgroup.fiu.edu/Economics/
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Related research

Keywords: Functional central limit theorem; extremal processes; tail empirical process; cadlag space; mixingale; near-epoch-dependence; regular variation; Hill estimator; tail dependence.;

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  1. Carrasco, Marine & Chen, Xiaohong, 2002. "Mixing And Moment Properties Of Various Garch And Stochastic Volatility Models," Econometric Theory, Cambridge University Press, vol. 18(01), pages 17-39, February.
  2. Hill, Jonathan B., 2010. "On Tail Index Estimation For Dependent, Heterogeneous Data," Econometric Theory, Cambridge University Press, vol. 26(05), pages 1398-1436, October.
  3. de Jong, Robert M. & Davidson, James, 2000. "The Functional Central Limit Theorem And Weak Convergence To Stochastic Integrals I," Econometric Theory, Cambridge University Press, vol. 16(05), pages 621-642, October.
  4. Quintos, Carmela & Fan, Zhenhong & Phillips, Peter C B, 2001. "Structural Change Tests in Tail Behaviour and the Asian Crisis," Review of Economic Studies, Wiley Blackwell, vol. 68(3), pages 633-63, July.
  5. Pham, Tuan D. & Tran, Lanh T., 1985. "Some mixing properties of time series models," Stochastic Processes and their Applications, Elsevier, vol. 19(2), pages 297-303, April.
  6. repec:cup:etheor:v:13:y:1997:i:3:p:353-67 is not listed on IDEAS
  7. Wooldridge, Jeffrey M. & White, Halbert, 1988. "Some Invariance Principles and Central Limit Theorems for Dependent Heterogeneous Processes," Econometric Theory, Cambridge University Press, vol. 4(02), pages 210-230, August.
  8. Robert M. De Jong & James Davidson, 2000. "Consistency of Kernel Estimators of Heteroscedastic and Autocorrelated Covariance Matrices," Econometrica, Econometric Society, vol. 68(2), pages 407-424, March.
  9. Einmahl, J.H.J., 1992. "Limit theorems for tail processes with application to intermediate quantile estimation," Open Access publications from Tilburg University urn:nbn:nl:ui:12-142067, Tilburg University.
  10. repec:fth:erroem:9024-a is not listed on IDEAS
  11. W. Breymann & A. Dias & P. Embrechts, 2003. "Dependence structures for multivariate high-frequency data in finance," Quantitative Finance, Taylor & Francis Journals, vol. 3(1), pages 1-14.
  12. Jonathan B. Hill, 2004. "Gaussian Tests of "Extremal White Noise" for Dependent, Heterogeneous, Heavy Tailed Time Series with an Application," Econometrics 0411014, EconWPA, revised 09 Dec 2004.
  13. de Jong, Robert M., 1997. "Central Limit Theorems for Dependent Heterogeneous Random Variables," Econometric Theory, Cambridge University Press, vol. 13(03), pages 353-367, June.
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