The Annals of Statistics

Estimating a tail exponent by modelling departure from a Pareto distribution

Andrey Feuerverger and Peter Hall

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We suggest two semiparametric methods for accommodating departures from a Pareto model when estimating a tail exponent by fitting the model to extreme-value data. The methods are based on approximate likelihood and on least squares, respectively. The latter is somewhat simpler to use and more robust against departures from classical extreme-value approximations, but produces estimators with approximately 64% greater variance when conventional extreme-value approximations are appropriate. Relative to the conventional assumption that the sampling population has exactly a Pareto distribution beyond a threshold, our methods reduce bias by an order of magnitude without inflating the order of variance. They are motivated by data on extrema of community sizes and are illustrated by an application in that context.

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Ann. Statist., Volume 27, Number 2 (1999), 760-781.

First available in Project Euclid: 5 April 2002

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Bias reduction extreme-value theory log-spacings maximum likelihood order statistics peaks-over-threshold regression regular variation spacings Zipf ’s law.


Feuerverger, Andrey; Hall, Peter. Estimating a tail exponent by modelling departure from a Pareto distribution. Ann. Statist. 27 (1999), no. 2, 760--781. doi:10.1214/aos/1018031215.

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  • CSORGO, S., DEHEUVELS, P. and MASON, D. 1985. Kernel estimates of the tail index of a ¨ distribution. Ann. Statist. 13 1050 1077. Z.
  • DAVID, H. A. 1970. Order Statistics. Wiley, New York. Z.
  • DAVISON, A. C. 1984. Modelling excesses over high thresholds, with an application. In StatistiZ. cal Extremes and Applications J. Tiago de Oliveira, ed. 461 482. Reidel, Dordrecht. Z.
  • DE HAAN, L. and RESNICK, S. I. 1980. A simple asymptotic estimate for the index of a stable distribution. J. Roy. Statist. Soc. Ser. B 42 83 88. Z.
  • HALL, P. 1982. On some simple estimates of an exponent of regular variation. J. Roy. Statist. Soc. Ser. B 44 37 42. Z.
  • HALL, P. 1990. Using the bootstrap to estimate mean squared error and select smoothing parameter in nonparametric problems. J. Multivariate Anal. 32 177 203. Z.
  • HALL, P. and WELSH, A. H. 1985. Adaptive estimates of parameters of regular variation. Ann. Statist. 13 331 341. Z.
  • HILL, B. M. 1970. Zipf 's law and prior distributions for the composition of a population. J. Amer. Statist. Assoc. 65 1220 1232. Z.
  • HILL, B. M. 1974. The rank frequency form of Zipf 's law. J. Amer. Statist. Assoc. 69 1017 1026. Z.
  • HILL, B. M. 1975. A simple general approach to inference about the tail of a distribution. Ann. Statist. 3 1163 1174. Z.
  • HILL, B. M. and WOODROOFE, M. 1975. Stronger forms of Zipf 's law. J. Amer. Statist. Assoc. 70 212 229. Z.
  • HOSKING, J. R. M. and WALLIS, J. R. 1987. Parameter and quantile estimation for the generalized Pareto distribution. Technometrics 29 339 349. Z.
  • HOSKING, J. R. M., WALLIS, J. R. and WOOD, E. F. 1985. Estimation of the generalized extreme-value distribution by the method of probability-weighted moments. Technometrics 27 251 261. Z. NERC 1975. Flood Studies Report 1. Natural Environment Research Council, London. Z.
  • PICKANDS, J. III 1975. Statistical inference using extreme order statistics. Ann. Statist. 3 119 131. Z.
  • ROOTZEN, H. and TAJVIDI, N. 1997. Extreme value statistics and wind storm losses: a case study. Scand. Actuar. J. 70 94. Z.
  • SMITH, R. L. 1984. Threshold methods for sample extremes. In Statistical Extremes and Z. Applications J. Tiago de Oliveira, ed. 621 638. Reidel, Dordrecht. Z.
  • SMITH, R. L. 1985. Maximum likelihood estimation in a class of nonregular cases. Biometrika 72 67 90. Z.
  • SMITH, R. L. 1989. Extreme value analysis of environmental time series: an application to trend detection in ground-level ozone. Statist. Sci. 4 367 393. Z.
  • TEUGELS, J. L. 1981. Limit theorems on order statistics. Ann. Probab. 9 868 880. Z.
  • TODOROVIC, P. 1978. Stochastic models of floods. Water Resource Research 14 345 356. Z.
  • WEISSMAN, I. 1978. Estimation of parameters and large quantiles based on the k largest observations. J. Amer. Statist. Assoc. 73 812 815.
  • WICKENS, C. H. 1921. Census of the Commonwealth of Australia, 1921 1. H. J. Green, Government Printer, Melbourne. Z.
  • ZIPF, G. K. 1941. National Unity and Disunity: the Nation as a Bio-Social Organism. Principia Press, Bloomington, IN. Z.
  • ZIPF, G. K. 1949. Human Behavior and the Principle of Least Effort: an Introduction to Human Ecology. Addison-Wesley, Cambridge, MA.