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Weighted Moment Estimators for the Second Order Scale Parameter

Author

Listed:
  • Tertius de Wet

    (University of Stellenbosch)

  • Yuri Goegebeur

    (University of Southern Denmark)

  • Armelle Guillou

    (Université de Strasbourg et CNRS)

Abstract

We consider the estimation of the scale parameter appearing in the second order condition when the distribution underlying the data is of Pareto-type. Inspired by the work of Goegebeur et al. (J Stat Plan Inference 140:2632–2652, 2010) on the estimation of the second order rate parameter, we introduce a flexible class of estimators for the second order scale parameter, which has weighted sums of scaled log spacings of successive order statistics as basic building blocks. Under the second order condition, some conditions on the weight functions, and for appropriately chosen sequences of intermediate order statistics, we establish the consistency of our class of estimators. Asymptotic normality is achieved under a further condition on the tail function 1 − F, the so-called third order condition. As the proposed estimator depends on the second order rate parameter, we also examine the effect of replacing the latter by a consistent estimator. The asymptotic performance of some specific examples of our proposed class of estimators is illustrated numerically, and their finite sample behavior is examined by a small simulation experiment.

Suggested Citation

  • Tertius de Wet & Yuri Goegebeur & Armelle Guillou, 2012. "Weighted Moment Estimators for the Second Order Scale Parameter," Methodology and Computing in Applied Probability, Springer, vol. 14(3), pages 753-783, September.
    • Handle: RePEc:spr:metcap:v:14:y:2012:i:3:d:10.1007_s11009-011-9263-6
    DOI: 10.1007/s11009-011-9263-6
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    References listed on IDEAS

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    1. Gomes, M. Ivette & Pestana, Dinis & Caeiro, Frederico, 2009. "A note on the asymptotic variance at optimal levels of a bias-corrected Hill estimator," Statistics & Probability Letters, Elsevier, vol. 79(3), pages 295-303, February.
    2. Holger Drees, 1998. "On Smooth Statistical Tail Functionals," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 25(1), pages 187-210, March.
    3. M. Ivette Gomes & Laurens De Haan & Lígia Henriques Rodrigues, 2008. "Tail index estimation for heavy‐tailed models: accommodation of bias in weighted log‐excesses," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 70(1), pages 31-52, February.
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