Summary
Denote byD(n, k) the maximum distance between the distribution function of thekth-largest order statistic in aniid sample of sizen, equally standardized ink, and of its corresponding limiting extreme value distribution.
Suppose that the underlying distribution function is ultimately continuous and strictly increasing. It is shown in the present paper thatD(n,k) converges to zero for any sequencek=k (n) with\(\mathop {\lim }\limits_{n \to \infty } k/n = 0\) if and only if the underlying distribution is ultimately a generalized Pareto distribution. ThusgPds do not only yield the best rate of joint convergence of extremes, but they are also the only distributions where there actually is convergence.
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Falk, M. A note on generalized Pareto distributions and thek upper extremes. Probab. Th. Rel. Fields 85, 499–503 (1990). https://doi.org/10.1007/BF01203167
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DOI: https://doi.org/10.1007/BF01203167