A goodness-of-fit statistic for Pareto-type behaviour

  • Authors:

  • Jan Beirlant;Tertius de Wet;Yuri Goegebeur

  • Affiliations:

  • Department of Mathematics, K.U. Leuven, Heverlee, Belgium;Department of Statistics and Actuarial Science, University of Stellenbosch, Matieland, South Africa;Department of Applied Economics, K.U. Leuven, Leuven, Belgium and University Centre for Statistics, K.U. Leuven, Heverlee, Belgium

  • Venue:
  • Journal of Computational and Applied Mathematics - Special issue: Jef Teugels
  • Year:
  • 2006

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Abstract

The fit of a statistical model can be visually assessed by inspection of a quantile-quantile or QQ plot. For the strict Pareto distribution, since log-transformed Pareto random variables are exponentially distributed, it is natural to consider an exponential quantile plot based on the log-transformed data. In case the data originate from a Pareto-type distribution, the Pareto quantile plot will be linear but only in some of the largest observations. In this paper we modify the Jackson statistic, originally proposed as a goodness-of-fit statistic for testing exponentiality, in such a way that it measures the linearity of the k largest observations on the Pareto quantile plot. Further, by taking the second-order tail behaviour of a Pareto-type model into account we construct a bias-corrected Jackson statistic. For both statistics the limiting distribution is derived. Next to these asymptotic results we also evaluate the small sample behaviour on the basis of a simulation study. The method is illustrated on two practical case studies.