, 12:361 | Cite as

Second order properties of distribution tails and estimation of tail exponents in random difference equations

  • Changryong Baek
  • Vladas Pipiras
  • Herwig Wendt
  • Patrice Abry


According to a celebrated result of Kesten (Acta Math 131:207–248, 1973), random difference equations have a power-law distribution tail in the asymptotic sense. Empirical evidence shows that classical estimators of tail exponent of random difference equations, such as Hill estimator, are extremely biased for larger values of tail exponents. It is argued in this work that the bias occurs because the power-tail region is too far in the tail from a practical perspective. This is supported by analysis of a few examples where a stationary distribution of random difference equation is known explicitly, and by proving a weaker form of the so-called second order regular variation of distribution tails of random difference equations, which measures deviations from the asymptotic power tail. The latter, in particular, suggests a specific second order term for a distribution tail. Estimation of tail exponents can be adapted by taking this second order term into account. One such method available in the literature is examined, and a new, simple, regression type estimator is proposed. Simulation study shows that the proposed estimator works very well. ARCH models of interest in Finance and multiplicative cascades used in Physics are considered as motivating examples throughout the work. Extension to multidimensional random difference equations with nonnegative entries is also considered.


Random difference equations Tail exponent and its estimation Second order regular variation ARCH models Multiplicative cascades 

AMS 2000 Subject Classifications

Primary—60G70 60H25 


  1. Baek, C., Pipiras, V.: Estimation of parameters in heavy-tailed distribution when its second order tail exponent is known. Preprint. Available at (2009)
  2. Basrak, B., Davis, R.A., Mikosch, T.: A characterization of multivariate regular variation. Ann. Appl. Probab. 12(3), 908–920 (2002a)MATHCrossRefMathSciNetGoogle Scholar
  3. Basrak, B., Davis, R.A., Mikosch, T.: Regular variation of GARCH processes. Stoch. Process. Appl. 99(1), 95–115 (2002b)MATHCrossRefMathSciNetGoogle Scholar
  4. Beirlant, J., Dierckx, G., Goegebeur, Y., Matthys, G.: Tail index estimation and an exponential regression model. Extremes 2(2), 177–200 (1999)MATHCrossRefMathSciNetGoogle Scholar
  5. Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular variation. In: Encyclopedia of Mathematics and its Applications, vol. 27. Cambridge University Press, Cambridge (1989)Google Scholar
  6. Bollerslev, T.: Generalized autoregressive conditional heteroskedasticity. J. Econom. 31(3), 307–327 (1986)MATHCrossRefMathSciNetGoogle Scholar
  7. Breiman, L.: On some limit theorems similar to the arc-sin law. Theory Probab. Appl. 10, 351–360 (1965)MATHCrossRefMathSciNetGoogle Scholar
  8. Chamayou, J.-F., Letac, G.: Explicit stationary distributions for compositions of random functions and products of random matrices. J. Theoret. Probab. 4(1), 3–36 (1991)MATHCrossRefMathSciNetGoogle Scholar
  9. de Haan, L., Ferreira, A.: Extreme Value Theory. Springer Series in Operations Research and Financial Engineering. Springer, New York (2006)MATHGoogle Scholar
  10. de Saporta, B., Guivarc’h, Y., Le Page, E.: On the multidimensional stochastic equation Y n + 1 = A n Y n + B n. Comptes Rendus Mathématique. Académie des Sciences. Paris 339(7), 499–502 (2004)MATHGoogle Scholar
  11. Embrechts, P., Klüppelberg, C., Mikosch, T.: Modelling Extremal Events. Springer, Berlin (1997)MATHGoogle Scholar
  12. Feuerverger, A., Hall, P.: Estimating a tail exponent by modelling departure from a Pareto distribution. Ann. Stat. 27(2), 760–781 (1999)MATHCrossRefMathSciNetGoogle Scholar
  13. Goldie, C.M.: Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Probab. 1(1), 126–166 (1991)MATHCrossRefMathSciNetGoogle Scholar
  14. Gomes, M., Ivette, d. H.L., Rodrigues, L.H.: Tail index estimation for heavy-tailed models: accommodation of bias in weighted log-excesses. J. R. Stat. Soc. Ser. B 70(Part 1), 31–52 (2008)MATHMathSciNetGoogle Scholar
  15. Grintsyavichyus, A.: On a random difference equation. Lith. Math. J. 21(4), 302–305 (1981)CrossRefMathSciNetGoogle Scholar
  16. Guivarc’h, Y.: Sur une extension de la notion de loi semi-stable. Ann. Inst. Henri Poincaré B Calc. Probab. Stat. 26(2), 261–285 (1990)MATHMathSciNetGoogle Scholar
  17. Guivarc’h, Y.: Heavy tail properties of stationary solutions of multidimensional stochastic recursions. In: IMS Lecture Notes Monogr. Ser., vol. 48, pp. 85–99. Institute of Mathematical Statistics, Beachwood (2006)Google Scholar
  18. Kahane, J.-P., Peyrière, J.: Sur certaines martingales de Benoit Mandelbrot. Adv. Math. 22(2), 131–145 (1976)MATHCrossRefGoogle Scholar
  19. Kesten, H.: Random difference equations and renewal theory for products of random matrices. Acta Math. 131, 207–248 (1973)MATHCrossRefMathSciNetGoogle Scholar
  20. Klüppelberg, C., Pergamenchtchikov, S.: The tail of the stationary distribution of a random coefficient AR(q) model. Ann. Appl. Probab. 14(2), 971–1005 (2004)MATHCrossRefMathSciNetGoogle Scholar
  21. Liu, Q.: On generalized multiplicative cascades. Stoch. Process. Appl. 86(2), 263–286 (2000)MATHCrossRefGoogle Scholar
  22. Mandelbort, B.: Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier. J. Fluid Mech. 62(2), 331–358 (1974)CrossRefGoogle Scholar
  23. Ossiander, M., Waymire, E.C.: Statistical estimation for multiplicative cascades. Ann. Stat. 28(6), 1533–1560 (2000)MATHCrossRefMathSciNetGoogle Scholar
  24. Peng, L.: Asymptotically unbiased estimators for the extreme-value index. Stat. Probab. Lett. 38(2), 107–115 (1998)CrossRefGoogle Scholar
  25. Resnick, S.I.: Heavy tail modeling and teletraffic data. Ann. Stat. 25(5), 1805–1869 (1997) (with discussion and a rejoinder by the author)MATHCrossRefMathSciNetGoogle Scholar
  26. Stelzer, R.: On Markov-switching ARMA processes—stationarity, existence of moments and geometric ergodicity. Econom. Theory 25, 43–62 (2009)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Changryong Baek
    • 1
  • Vladas Pipiras
    • 1
  • Herwig Wendt
    • 2
  • Patrice Abry
    • 2
  1. 1.Department of Statistics and Operations ResearchUNC at Chapel HillChapel HillUSA
  2. 2.Laboratoire de PhysiqueEcole Normale Supérieure de LyonLyon cedex 7France

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