Abstract
Let M nr be the rth largest of a random sample of size n from a distribution F of exponential power type on R. That is, 1-F(z) = O(x d exp(−x)) as x = (z/σ)α → ∞. For example, the exponential, gamma, chi-square, Laplace and normal distributions are of this type. We obtain an asymptotic expansion in powers of u 1 = −log(1 − u) and u 2 = log u 1, for the quantile F −1(u) near u = 1. From this, we obtain a double expansion in inverse powers of (log n, n) for the moments of M nr /n 1/n 1/α, with the coefficient a polynomial in log log n. We also discuss a possible application to an optimal stopping problem.
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Withers, C.S., Nadarajah, S. Expansions for quantiles and moments of extremes for distributions of exponential power type. Sankhya A 73, 202–217 (2011). https://doi.org/10.1007/s13171-011-0014-0
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DOI: https://doi.org/10.1007/s13171-011-0014-0