Logarithmic Quantile Estimation for Rank Statistics

• Activity Number: 532
• Type: Contributed
• Date/Time: Wednesday, August 7, 2013 : 10:30 AM to 12:20 PM
• Sponsor: Section on Nonparametric Statistics
• Author(s): Lucia Tabacu*+ and Manfred Denker
• Companies: Pennsylvania State University and Mathematics Department, Pennsylvania State University
• Keywords: logarithmic quantile estimation ; rank statistics ; nonparametric designs ; almost sure weak convergence ; quadratic forms ; hypothesis testing; data analysis

Abstract

We introduce a new estimation procedure for quantiles in nonparametric factorial designs. This is based on a new average procedure, known as almost sure central limit theorem. We state such a limit theorem for simple linear rank statistics for samples with continuous distribution functions. As a corollary the result extends to samples with ties, and the vector version of an almost sure 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 and we confirm this finding for the c-sample problem. We finally demonstrate the method using the 'shoulder tip pain' study in Lumley, 1996.