Skip to main content Accessibility help
×
Home

Estimating tail decay for stationary sequences via extreme values

  • Assaf Zeevi (a1) and Peter W. Glynn (a2)

Abstract

We study estimation of the tail-decay parameter of the marginal distribution corresponding to a discrete-time, real-valued stationary stochastic process. Assuming that the underlying process is short-range dependent, we investigate properties of estimators of the tail-decay parameter which are based on the maximal extreme value of the process observed over a sampled time interval. These estimators only assume that the tail of the marginal distribution is roughly exponential, plus some modest ‘mixing’ conditions. Consistency properties of these estimators are established, as well as minimax convergence rates. We also provide some discussion on estimating the pre-exponent, when a more refined tail asymptotic is assumed. Properties of a certain moving-average variant of the extremal-based estimator are investigated as well. In passing, we also characterize the precise dependence (mixing) assumptions that support almost-sure limit theory for normalized extreme values and related first-passage times in stationary sequences.

Copyright

Corresponding author

Postal address: Graduate School of Business, Columbia University, 3022 Broadway, New York, NY 10027, USA. Email address: assaf@gsb.columbia.edu
∗∗ Postal address: Management Science and Engineering, Terman Engineering Center, Stanford University, Stanford, CA 94305, USA.

References

Hide All
[1] Asmussen, S. (1998). Extreme value theory for queues via cycle maxima. Extremes 1, 137168.
[2] Athreya, K. B. and Pantula, S. G. (1986). Mixing properties of Harris chains and autoregressive processes. J. Appl. Prob. 23, 880892.
[3] Beran, J., Sherman, R., Taqqu, M. and Willinger, W. (1995). Long-range dependence in variable-bit-rate video traffic. IEEE Trans. Commun. 43, 15661579.
[4] Berger, A. W. and Whitt, W. (1995). Maximum values in queueing processes. Prob. Eng. Inf. Sci. 9, 375409.
[5] Berman, S. M. (1964). Limit theorems for the maximum term in stationary sequences. Ann. Math. Statist. 35, 502516.
[6] Bertsimas, D. and Paschalidis, I. (2001). Probabilistic service level guarantees in make-to-stock manufacturing systems. Operat. Res. 49, 119133.
[7] Bertsimas, D., Paschalidis, I. and Tsitsiklis, J. N. (1998). On the large deviations behaviour of acyclic networks of G/G/1 queues. Ann. Appl. Prob. 8, 10271069.
[8] Billingsley, P. (1986). Probability and Measure. John Wiley, New York.
[9] Bradley, R. C. (1986). Basic properties of strong mixing conditions. In Dependence in Probability and Statistics (Progr. Prob. Statist. 11), Birkhäuser, Boston, MA, pp. 165192.
[10] Courcoubetis, C. et al. (1995). Admission control and routing in ATM networks using inference from measured buffer occupancy. IEEE Trans. Commun. 43, 17781784.
[11] Devroye, L., Györfi, L. and Lugosi, G. (1996). A Probabilistic Theory of Pattern Recognition. Springer, New York.
[12] Doukhan, P. (1994). Mixing. Properties and Examples. Springer, New York.
[13] Drees, H. (2001). Minimax risk bounds in extreme value theory. Ann. Statist. 21, 266294.
[14] Duffield, N. G. and O'Connell, N. (1995). Large deviations and overflow probabilities with general single-server queue, with applications. Math. Proc. Camb. Phil. Soc. 118, 363374.
[15] Embrechts, P., Klüppelberg, T. and Mikosch, C. (1997). Modelling Extremal Events. Springer, Berlin.
[16] Galambos, J. (1978). The Asymptotic Theory of Extreme Order Statistics. John Wiley, New York.
[17] Glasserman, P. and Kou, S. (1995). Limits of first passage times to rare sets in regenerative processes. Ann. Appl. Prob. 5, 424445.
[18] Glynn, P. W. (1982). Simulation output analysis for general state space Markov chains. Doctoral Thesis, Stanford University.
[19] Glynn, P. W. and Whitt, W. (1994). Logarithmic asymptotics for steady-state tail probabilities in a single-server queue. In Studies in Applied Probability (J. Appl. Prob. Spec. Vol. 31A), eds Galambos, J. and Gani, J., Applied Probability Trust, Sheffield, pp. 131156.
[20] Glynn, P. W. and Zeevi, A. J. (2000). Estimating tail probabilities in queues via extremal statistics. In Analysis of Communication Networks: Call Centres, Traffic and Performance (Fields Inst. Commun. 28), American Mathematical Society, Providence, RI, pp. 135158.
[21] Haiman, G. and Habach, L. (1999). Almost sure behaviour of extremes of m-dependent stationary sequences. C. R. Acad. Sci. Paris Sér. I Math. 329, 887892.
[22] Hall, P., Teugels, J. L. and Vanmarcke, A. (1992). The abscissa of convergence of the Laplace transform. J. Appl. Prob. 29, 353362.
[23] Hsu, I. and Walrand, J. (1996). Dynamic bandwidth allocation for ATM switches. J. Appl. Prob. 33, 758771.
[24] Iglehart, D. L. (1972). Extreme values in the GI/G/1 queue. Ann. Math. Statist. 43, 627635.
[25] Kesten, H. and O'Brien, G. L. (1976). Examples of mixing sequences. Duke Math. J. 43, 405415.
[26] Leadbetter, M. R., Lindgren, G. and Rootzén, H. (1983). Extremes and Related Properties of Random Sequences and Processes. Springer, New York.
[27] Loynes, R. M. (1965). Extreme values in uniformly mixing stationary stochastic processes. Ann. Math. Statist. 36, 993999.
[28] Mokkadem, A. (1988). Mixing properties of ARMA processes. Stoch. Process. Appl. 29, 309315.
[29] Nobel, A. B. and Dembo, A. (1993). A note on uniform laws of averages for dependent processes. Statist. Prob. Lett. 17, 169172.
[30] O'Brien, G. L. (1987). Extreme values for stationary and Markov sequences. Ann. Prob. 15, 281291.
[31] Paschalidis, I. and Vassilaras, S. (2001). On the estimation of buffer overflow probabilities from measurements. IEEE Trans. Inf. Theory 47, 178191.
[32] Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York.
[33] Rootzén, H., (1988). Maxima and exceedances of stationary Markov chains. Adv. Appl. Prob. 20, 371390.
[34] Zeevi, A. and Glynn, P. W. (2000). On the maximum workload of a queue fed by fractional Brownian motion. Ann. Appl. Prob. 10, 10841099.
[35] Zeevi, A. and Glynn, P. W. (2001). Estimating tail decay for stationary sequences via extreme values. Tech. Rep., Graduate School of Business, Columbia University.

Keywords

MSC classification

Estimating tail decay for stationary sequences via extreme values

  • Assaf Zeevi (a1) and Peter W. Glynn (a2)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed