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.
Extreme value statistics Pareto-type model Second order scale parameter Weighted estimator
AMS 2000 Subject Classifications
62G05 62G20 62G30 62G32
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Goegebeur Y, de Wet T (2011) Estimation of the third order parameter in extreme value statistics. doi:10.1007/s11749-011-0246-2 (to appear in TEST)
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