Logarithmic Quantile Estimation for Rank Statistics

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 the vector version of an 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 (\cite{THA, FRI}) and we confirm this finding for the c-sample problem

Testing Longitudinal Data by Logarithmic Quantiles

The shoulder tip pain study of Lumley [13] is re-investigated. It is shown that the new logarithmic quantile estimation (LQE) technique in [9] applies and behaves well under singular covariance structure and small sample sizes as in the shoulder tip pain study. The findings in [6] can be assured under weaker assumptions using a combination of LQE and an ANOVA type statistic. © 2014, Institute of Mathematical Statistics