On the estimation of the heavy-tail exponent in time series using the max-spectrum

This paper addresses the problem of estimating the tail index α of distributions with heavy, Pareto-type tails for dependent data, that is of interest in the areas of finance, insurance, environmental monitoring and teletraffic analysis. A novel approach based on the max self-similarity scaling behavior of block maxima is introduced. The method exploits the increasing lack of dependence of maxima over large size blocks, which proves useful for time series data. We establish the consistency and asymptotic normality of the proposed max-spectrum estimator for a large class of m-dependent time series, in the regime of intermediate block-maxima. In the regime of large block-maxima, we demonstrate the distributional consistency of the estimator for a broad range of time series models including linear processes. The max-spectrum estimator is a robust and computationally efficient tool, which provides a novel time-scale perspective to the estimation of the tail exponents. Its performance is illustrated over synthetic and real data sets. Copyright © 2009 John Wiley & Sons, Ltd.

[1]  Jonathan A. Tawn,et al.  A Comparison of Methods for Estimating the Extremal Index , 2000 .

[2]  B. Mandelbrot.  THE PARETO-LEVY LAW AND THE DISTRIBUTION OF INCOME* , 1960 .

[3]  M. Crovella,et al.  Heavy-tailed probability distributions in the World Wide Web , 1998 .

[4]  Paul Deheuvels,et al.  Kernel Estimates of the Tail Index of a Distribution , 1985 .

[5]  Liang Peng,et al.  Likelihood Based Confidence Intervals for the Tail Index , 2002 .

[6]  S. Resnick.  Heavy tail modeling and teletraffic data: special invited paper , 1997 .

[7]  L. de Haan,et al.  A Spectral Representation for Max-stable Processes , 1984 .

[8]  Michalis Faloutsos,et al.  On power-law relationships of the Internet topology , 1999, SIGCOMM '99.

[9]  Paul Deheuvels,et al.  Point processes and multivariate extreme values , 1983 .

[10]  Prabhakar Raghavan,et al.  Computing on data streams , 1999, External Memory Algorithms.

[11]  R. Adler,et al.  A practical guide to heavy tails: statistical techniques and applications , 1998 .

[12]  Lada A. Adamic,et al.  The Nature of Markets in the World Wide Web , 1999 .

[13]  Azer Bestavros,et al.  Self-similarity in World Wide Web traffic: evidence and possible causes , 1997, TNET.

[14]  Ward Whitt,et al.  An Introduction to Stochastic-Process Limits and their Application to Queues , 2002 .

[15]  Walter Willinger,et al.  Self-Similar Network Traffic and Performance Evaluation , 2000 .

[16]  Sidney Resnick,et al.  Smoothing the Hill Estimator , 1997, Advances in Applied Probability.

[17]  B. M. Hill,et al.  A Simple General Approach to Inference About the Tail of a Distribution , 1975 .

[18]  Sidney I. Resnick,et al.  Limit Theory for Moving Averages of Random Variables with Regularly Varying Tail Probabilities , 1985 .

[19]  S. Resnick.  Extreme Values, Regular Variation, and Point Processes , 1987 .

[20]  Sidney I. Resnick,et al.  How to make a Hill Plot , 2000 .

[21]  Ronald L. Rivest,et al.  Introduction to Algorithms , 1990 .

[22]  Richard L. Smith,et al.  The behavior of multivariate maxima of moving maxima processes , 2004 .

[23]  Sidney I. Resnick,et al.  Prediction of Stationary Max-Stable Processes , 1993 .

[24]  S. Resnick,et al.  The qq-estimator and heavy tails , 1996 .

[25]  Murad S. Taqqu,et al.  Extremal stochastic integrals: a parallel between max-stable processes and α-stable processes , 2005 .

[26]  Lada A. Adamic,et al.  Zipf's law and the Internet , 2002, Glottometrics.

[27]  George Michailidis,et al.  Estimating Heavy-Tail Exponents Through Max Self–Similarity , 2006, IEEE Transactions on Information Theory.

[28]  Jiang Wang,et al.  Trading and Returns under Periodic Market Closures , 2000 .

[29]  Albert-László Barabási,et al.  Linked: The New Science of Networks , 2002 .

[30]  P. Hall.  On Some Simple Estimates of an Exponent of Regular Variation , 1982 .

[31]  S. Muthukrishnan,et al.  Surfing Wavelets on Streams: One-Pass Summaries for Approximate Aggregate Queries , 2001, VLDB.

[32]  George Michailidis,et al.  A Diagnostic Plot for Estimating the Tail Index of a Distribution , 2004 .

[33]  L. Peng,et al.  Confidence intervals for the tail index , 2001 .

[34]  S. Muthukrishnan,et al.  Data streams: algorithms and applications , 2005, SODA '03.

[35]  A. V. D. Vaart,et al.  Asymptotic Statistics: Frontmatter , 1998 .

[36]  Holger Rootzén,et al.  Extreme Values in Finance, Telecommunications, and the Environment , 2003 .

[37]  Jiang Wang,et al.  Trading Volume: Definitions, Data Analysis, and Implications of Portfolio Theory , 2000 .

[38]  Walter Willinger,et al.  Proof of a fundamental result in self-similar traffic modeling , 1997, CCRV.

[39]  M. R. Leadbetter,et al.  Extremes and Related Properties of Random Sequences and Processes: Springer Series in Statistics , 1983 .

[40]  Kihong Park,et al.  On the relationship between file sizes, transport protocols, and self-similar network traffic , 1996, Proceedings of 1996 International Conference on Network Protocols (ICNP-96).

[41]  M. Crovella,et al.  Estimating the Heavy Tail Index from Scaling Properties , 1999 .

[42]  Sidney I. Resnick,et al.  Heavy Tail Modelling and Teletraffic Data , 1995 .

[43]  S. Resnick,et al.  Consistency of Hill's estimator for dependent data , 1995, Journal of Applied Probability.

[44]  Gennady Samorodnitsky,et al.  The supremum of a negative drift random walk with dependent heavy-tailed steps , 2000 .

[45]  J M Carlson,et al.  Highly optimized tolerance: a mechanism for power laws in designed systems. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[46]  Ruey S. Tsay,et al.  Analysis of Financial Time Series , 2005 .

[47]  P. Hall,et al.  Estimating a tail exponent by modelling departure from a Pareto distribution , 1999 .