Linking Pareto-Tail Kernel Goodness-Offit Statistics with Tail Index at Optimal Threshold and Second Order Estimation

Abstract

In this paper the relation between goodness-of-fit testing and the optimal selection of the sample fraction for tail estimation, for instance using Hill’s estimator, is examined. We consider this problem under a general kernel goodness-of-fit test statistic for assessing whether a sample is consistent with the Pareto-type model. The derivation of the class of kernel goodness-of-fit statistics is based on the close link between the strict Pareto and the exponential distribution, and puts some of the available goodness-offit procedures for the latter in a broader perspective. Two important special cases of the kernel statistic, the Jackson and the Lewis statistic, will be discussed in greater depth. The relationship between the limiting distribution of the Lewis statistic and the bias-component of the asymptotic mean squared error of the Hill estimator is exploited to construct a new tail sample fraction selection criterion for the latter. The methodology is illustrated on a case study.