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In samples from a heavy-tailed distribution a second-order approximation isoften use to approximate the tail function. Based on the parameters of theapproximation, an optimal sample fraction can be estimated which is then usedto estimate the index. Given that the observations are above a threshold andhas an approximate generalized Pareto distribution, an expression is derivedfor the percentile above which the second-order approximation is valid

The sample fraction in peaks-over-threshold problems where the  second-order expansion is valid with specific reference to the generalized  Pareto distribution