The Weighted Bootstrap Mean for Heavy-Tailed Distributions |
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Authors: | E del Barrio C Matrán |
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Institution: | (1) Departamento de Estadística e Investigación Operativa, Universidad de Valladolid, Spain;(2) Departamento de Estadística e Investigación Operativa, Universidad de Valladolid, Spain |
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Abstract: | We study the performance of the weighted bootstrap of the mean of i.i.d. random variables, X
1, X
2,..., in the domain of attraction of an -stable law, 1< <2. In agreement with the results, in the Efron's bootstrap setup, by Athreya,(4) Arcones and Giné(2) and Deheuvels et al.,(11) we prove that for a low resampling intensity the weighted bootstrap works in probability. Our proof resorts to the 0–1 law methodology introduced in Arenal and Matrán(3). This alternative to the methodology initiated in Mason and Newton(25) presents the advantage that it does not use Hájek's Central Limit Theorem for linear rank statistics which actually only provides normal limit laws. We include as an appendix a sketched proof, based on the Komlos–Major–Tusnady construction, of the asymptotic behaviour of the Wasserstein distance between the empirical and the parent distribution of a sample, which is also a main tool in our development. |
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Keywords: | weighted bootstrap heavy tailed distributions -stable law" target="_blank">gif" alt="agr" align="BASELINE" BORDER="0">-stable law domain of attraction resampling intensity regular variation Wasserstein distance |
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