Tail Behavior of Threshold Models with Innovations in the Domain of Attraction of the Double Exponential Distribution

ABSTRACT

We consider a two-regime threshold autoregressive model where the driving noises are sequences of independent and identically distributed random variables with common distribution function which belongs to the domain of attraction of double exponential distribution. If in addition, for each and where denotes the convolution of the distribution function and we determine the tail behavior of the process and give the exact values of the coefficient.

We consider a two-regime threshold autoregressive model where the driving noises are sequences of independent and identically distributed random variables with common distribution function which belongs to the domain of attraction of double exponential distribution. If in addition, for each and where denotes the convolution of the distribution function and we determine the tail behavior of the process and give the exact values of the coefficient.

KEYWORDS

Tail Behavior, Domain of Attraction, Convolution Tails, Stochastic Recurrence Equation, Threshold Autoregressive Model

Tail Behavior, Domain of Attraction, Convolution Tails, Stochastic Recurrence Equation, Threshold Autoregressive Model

Cite this paper

nullA. Diop and S. Diouf, "Tail Behavior of Threshold Models with Innovations in the Domain of Attraction of the Double Exponential Distribution,"*Applied Mathematics*, Vol. 2 No. 5, 2011, pp. 515-520. doi: 10.4236/am.2011.25067.

nullA. Diop and S. Diouf, "Tail Behavior of Threshold Models with Innovations in the Domain of Attraction of the Double Exponential Distribution,"

References

[1] H. Tong, “Non-Linear Time Series: A Dynamical System Approach,” Oxford University Press, New York, 1990.

[2] H. Kesten, “Random Difference Equations and Renewal Theory for Products of Random Matrices,” Acta Mathematica, Vol. 131, No. 4, 1973, pp. 207-248. doi:10.1007/BF02392040U

[3] R. A. Davis and S. I. Resnick, “Limit Theory for Bilinear Process with Heavy-Tailed,” The Annals of Applied Probability, Vol. 6, No. 4, 1996, pp. 1191-1210. Udoi:10.1214/aoap/1035463328U

[4] S. I. Resnick and E. Willekens, “Moving Averages with Random Coefficient Autoregressive Models,” Stochastic Models, Vol. 7, No. 4, 1991, pp. 511-525.

[5] A. Diop and D. Guégan, “Tail Behavior of Threshold Autoregressive Stochastic Volatility Model,” Extremes, Vol. 7,No. 4, 2004, pp. 367-375. doi:10.1007/s10687-004-3482-yU

[6] D. B. H. Cline, “Convolution Tails, Products Tails and Domains of Attraction,” Probability Theory and Related Fields, Vol. 72, No. 4, 1986, pp. 529-557. Udoi:10.1007/BF00344720U

[7] P. Embrechts, “Subexponential Distribution Functions and Their Applications: A Review,” Proceedings of the Seventh Conference Probability Theory, Brasov, 29 August-4 September 1982, pp. 125-136.

[8] C. Klüppelberg, “Subexponential Distributions and Integrated Tails,” Journal of Applied Probability, Vol. 25, No. 1, 1988, pp. 132-141. Udoi:10.2307/3214240U

[9] R. E. Beard, T. Pentikainen and E. Pesonen, “Risk Theory,” Chapman & Hall, London, 1984.

[10] R. V. Hogg and S. A. Klugman, “Loss Distributions,” Wiley, New York, 1984. doi:10.1002/9780470316634U

[11] C. M. Goldie and S. I. Resnick, “Distributions That Are Both Subexponential and in the Domain of Attraction of an Extreme-Value Distribution,” Advances in Applied Probability, Vol. 20, No. 4, 1988, pp. 706-718. Udoi:10.2307/1427356U

[12] A. Brandt, “The Stochastic Equation Yn+1 = AnYn + Bn with Stationary Coefficients,” Advances in Applied Probability, Vol. 18, No. 1, 1986, pp. 211-220. Udoi:10.2307/1427243U

[13] P. Embrechts, C. Kluppelberg and T. Mikosch, “Modelling Extremal Events for Insurance and Finance,” Springer Verlag, Berlin, 1997.

[14] F. J. Breidt, “A Threshold Autoregressive Stochastic Volatility Model,” VI Latin American Congress of Probability and Mathematical Statistics(CLAPEM), Valparaiso, 1996.

[15] H. Tong, “On a Threshold Model, Pattern Recognition and Signal Processing,” C. H. Chen, Ed., Sijhoff & Noordoff, Amsterdam, 1990.

[16] J. Gonzalo and R. Montesinos, “Threshold Stochastic Unit Root Models,” Manuscript, Universidad Carlos Ⅲ, 2002.

[17] C. Gouriéroux and C. Y. Robert, “Stochastic Unit Root Models,” Econometric Theory, Vol. 22, No. 6, 2006, pp. 1052-1090. doi: 10.1017/S0266466606060518

[18] R. Davis and S. Resnick “Extremes of Moving Averages of Random Variables from the Domain of Attraction of the Double Exponential Distribution,” Stochastic Processes and their Applications, Vol. 30, No. 1, 1988, pp. 41-68. doi:10.1016/0304-4149(88)90075-0U

[19] A. A. Balkema and L. De Haan, “On R. Von Mises’s Condition for the Domain of Attraction of Exp(–e–x),” The Annals of Mathematical Statistics, Vol. 43, No. 4, 1972, pp. 1352-1354. doi:10.1214/aoms/1177692489U

[1] H. Tong, “Non-Linear Time Series: A Dynamical System Approach,” Oxford University Press, New York, 1990.

[2] H. Kesten, “Random Difference Equations and Renewal Theory for Products of Random Matrices,” Acta Mathematica, Vol. 131, No. 4, 1973, pp. 207-248. doi:10.1007/BF02392040U

[3] R. A. Davis and S. I. Resnick, “Limit Theory for Bilinear Process with Heavy-Tailed,” The Annals of Applied Probability, Vol. 6, No. 4, 1996, pp. 1191-1210. Udoi:10.1214/aoap/1035463328U

[4] S. I. Resnick and E. Willekens, “Moving Averages with Random Coefficient Autoregressive Models,” Stochastic Models, Vol. 7, No. 4, 1991, pp. 511-525.

[5] A. Diop and D. Guégan, “Tail Behavior of Threshold Autoregressive Stochastic Volatility Model,” Extremes, Vol. 7,No. 4, 2004, pp. 367-375. doi:10.1007/s10687-004-3482-yU

[6] D. B. H. Cline, “Convolution Tails, Products Tails and Domains of Attraction,” Probability Theory and Related Fields, Vol. 72, No. 4, 1986, pp. 529-557. Udoi:10.1007/BF00344720U

[7] P. Embrechts, “Subexponential Distribution Functions and Their Applications: A Review,” Proceedings of the Seventh Conference Probability Theory, Brasov, 29 August-4 September 1982, pp. 125-136.

[8] C. Klüppelberg, “Subexponential Distributions and Integrated Tails,” Journal of Applied Probability, Vol. 25, No. 1, 1988, pp. 132-141. Udoi:10.2307/3214240U

[9] R. E. Beard, T. Pentikainen and E. Pesonen, “Risk Theory,” Chapman & Hall, London, 1984.

[10] R. V. Hogg and S. A. Klugman, “Loss Distributions,” Wiley, New York, 1984. doi:10.1002/9780470316634U

[11] C. M. Goldie and S. I. Resnick, “Distributions That Are Both Subexponential and in the Domain of Attraction of an Extreme-Value Distribution,” Advances in Applied Probability, Vol. 20, No. 4, 1988, pp. 706-718. Udoi:10.2307/1427356U

[12] A. Brandt, “The Stochastic Equation Yn+1 = AnYn + Bn with Stationary Coefficients,” Advances in Applied Probability, Vol. 18, No. 1, 1986, pp. 211-220. Udoi:10.2307/1427243U

[13] P. Embrechts, C. Kluppelberg and T. Mikosch, “Modelling Extremal Events for Insurance and Finance,” Springer Verlag, Berlin, 1997.

[14] F. J. Breidt, “A Threshold Autoregressive Stochastic Volatility Model,” VI Latin American Congress of Probability and Mathematical Statistics(CLAPEM), Valparaiso, 1996.

[15] H. Tong, “On a Threshold Model, Pattern Recognition and Signal Processing,” C. H. Chen, Ed., Sijhoff & Noordoff, Amsterdam, 1990.

[16] J. Gonzalo and R. Montesinos, “Threshold Stochastic Unit Root Models,” Manuscript, Universidad Carlos Ⅲ, 2002.

[17] C. Gouriéroux and C. Y. Robert, “Stochastic Unit Root Models,” Econometric Theory, Vol. 22, No. 6, 2006, pp. 1052-1090. doi: 10.1017/S0266466606060518

[18] R. Davis and S. Resnick “Extremes of Moving Averages of Random Variables from the Domain of Attraction of the Double Exponential Distribution,” Stochastic Processes and their Applications, Vol. 30, No. 1, 1988, pp. 41-68. doi:10.1016/0304-4149(88)90075-0U

[19] A. A. Balkema and L. De Haan, “On R. Von Mises’s Condition for the Domain of Attraction of Exp(–e–x),” The Annals of Mathematical Statistics, Vol. 43, No. 4, 1972, pp. 1352-1354. doi:10.1214/aoms/1177692489U