Hostname: page-component-84b7d79bbc-4hvwz Total loading time: 0 Render date: 2024-08-01T12:27:43.489Z Has data issue: false hasContentIssue false
  • English
  • Français

Estimating tail decay for stationary sequences via extreme values

Part of: Operations research and management science Stochastic processes Nonparametric inference

Published online by Cambridge University Press:  01 July 2016

Assaf Zeevi  and
Peter W. Glynn
Assaf Zeevi*
Affiliation:
Columbia University
Peter W. Glynn*
Affiliation:
Stanford University
*
∗ 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.
Get access Rights & Permissions [Opens in a new window]

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.

Keywords

Extreme valuesestimationtail behaviorstrong mixinguniform mixingminimaxalmost-sure convergence

MSC classification

Primary: 60G70: Extreme value theory; extremal processes
Secondary: 62G32: Statistics of extreme values; tail inference 90B20: Traffic problems
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 36 , Issue 1 , March 2004 , pp. 198 - 226
Copyright
Copyright © Applied Probability Trust 2004 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Asmussen, S. (1998). Extreme value theory for queues via cycle maxima. Extremes 1, 137–168.CrossRefGoogle Scholar
[2] Athreya, K. B. and Pantula, S. G. (1986). Mixing properties of Harris chains and autoregressive processes. J. Appl. Prob. 23, 880–892.Google Scholar
[3] Beran, J., Sherman, R., Taqqu, M. and Willinger, W. (1995). Long-range dependence in variable-bit-rate video traffic. IEEE Trans. Commun. 43, 1566–1579.Google Scholar
[4] Berger, A. W. and Whitt, W. (1995). Maximum values in queueing processes. Prob. Eng. Inf. Sci. 9, 375–409.CrossRefGoogle Scholar
[5] Berman, S. M. (1964). Limit theorems for the maximum term in stationary sequences. Ann. Math. Statist. 35, 502–516.CrossRefGoogle Scholar
[6] Bertsimas, D. and Paschalidis, I. (2001). Probabilistic service level guarantees in make-to-stock manufacturing systems. Operat. Res. 49, 119–133.CrossRefGoogle Scholar
[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, 1027–1069.Google Scholar
[8] Billingsley, P. (1986). Probability and Measure. John Wiley, New York.Google Scholar
[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. 165–192.CrossRefGoogle Scholar
[10] Courcoubetis, C. et al. (1995). Admission control and routing in ATM networks using inference from measured buffer occupancy. IEEE Trans. Commun. 43, 1778–1784.Google Scholar
[11] Devroye, L., Györfi, L. and Lugosi, G. (1996). A Probabilistic Theory of Pattern Recognition. Springer, New York.CrossRefGoogle Scholar
[12] Doukhan, P. (1994). Mixing. Properties and Examples. Springer, New York.Google Scholar
[13] Drees, H. (2001). Minimax risk bounds in extreme value theory. Ann. Statist. 21, 266–294.Google Scholar
[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, 363–374.CrossRefGoogle Scholar
[15] Embrechts, P., Klüppelberg, T. and Mikosch, C. (1997). Modelling Extremal Events. Springer, Berlin.Google Scholar
[16] Galambos, J. (1978). The Asymptotic Theory of Extreme Order Statistics. John Wiley, New York.Google Scholar
[17] Glasserman, P. and Kou, S. (1995). Limits of first passage times to rare sets in regenerative processes. Ann. Appl. Prob. 5, 424–445.Google Scholar
[18] Glynn, P. W. (1982). Simulation output analysis for general state space Markov chains. Doctoral Thesis, Stanford University.Google Scholar
[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. 131–156.Google Scholar
[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. 135–158.Google Scholar
[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, 887–892.CrossRefGoogle Scholar
[22] Hall, P., Teugels, J. L. and Vanmarcke, A. (1992). The abscissa of convergence of the Laplace transform. J. Appl. Prob. 29, 353–362.Google Scholar
[23] Hsu, I. and Walrand, J. (1996). Dynamic bandwidth allocation for ATM switches. J. Appl. Prob. 33, 758–771.Google Scholar
[24] Iglehart, D. L. (1972). Extreme values in the GI/G/1 queue. Ann. Math. Statist. 43, 627–635.Google Scholar
[25] Kesten, H. and O'Brien, G. L. (1976). Examples of mixing sequences. Duke Math. J. 43, 405–415.Google Scholar
[26] Leadbetter, M. R., Lindgren, G. and Rootzén, H. (1983). Extremes and Related Properties of Random Sequences and Processes. Springer, New York.Google Scholar
[27] Loynes, R. M. (1965). Extreme values in uniformly mixing stationary stochastic processes. Ann. Math. Statist. 36, 993–999.Google Scholar
[28] Mokkadem, A. (1988). Mixing properties of ARMA processes. Stoch. Process. Appl. 29, 309–315.Google Scholar
[29] Nobel, A. B. and Dembo, A. (1993). A note on uniform laws of averages for dependent processes. Statist. Prob. Lett. 17, 169–172.Google Scholar
[30] O'Brien, G. L. (1987). Extreme values for stationary and Markov sequences. Ann. Prob. 15, 281–291.Google Scholar
[31] Paschalidis, I. and Vassilaras, S. (2001). On the estimation of buffer overflow probabilities from measurements. IEEE Trans. Inf. Theory 47, 178–191.CrossRefGoogle Scholar
[32] Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York.CrossRefGoogle Scholar
[33] Rootzén, H., (1988). Maxima and exceedances of stationary Markov chains. Adv. Appl. Prob. 20, 371–390.Google Scholar
[34] Zeevi, A. and Glynn, P. W. (2000). On the maximum workload of a queue fed by fractional Brownian motion. Ann. Appl. Prob. 10, 1084–1099.Google Scholar
[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.Google Scholar