Logarithmic Quantile Estimation for Rank Statistics

Abstract

We prove an almost sure weak limit theorem for simple linear rank statistics for samples with continuous distributions functions. As a corollary, the result extends to samples with ties and to the vector version of an almost sure (a.s.) central limit theorem for vectors of linear rank statistics. Moreover, we derive such a weak convergence result for some quadratic forms. These results are then applied to quantile estimation, and to hypothesis testing for nonparametric statistical designs, here demonstrated by the c-sample problem, where the samples may be dependent. In general, the method is known to be comparable to the bootstrap and other nonparametric methods (Thangavelu 2005; Fridline 2009), and we confirm this finding for the c-sample problem.

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Change history

  • 01 March 2017

    The main theorem was incorrect in the original online and print publications. The condition in Eq. (11) within Theorem 2.1 should read

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Correspondence to Lucia Tabacu.

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Denker, M., Tabacu, L. Logarithmic Quantile Estimation for Rank Statistics. J Stat Theory Pract 9, 146–170 (2015). https://doi.org/10.1080/15598608.2014.886312

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AMS Subject Classification

  • 62E20
  • 62G20

Keywords

  • Almost sure central limit theorem
  • Rank statistics
  • Logarithmic quantile estimation
  • Kruskal-Wallis statistic