Logarithmic Quantile Estimation for Rank Statistics

TL;DR

This paper establishes almost sure weak limit theorems for linear and quadratic rank statistics, extending to ties and dependent samples, with applications to quantile estimation and nonparametric hypothesis testing.

Contribution

It introduces a new almost sure weak limit theorem for rank statistics, including cases with ties and dependence, enhancing nonparametric statistical methods.

Findings

Theorem extends to samples with ties.

Results apply to dependent samples.

Method is comparable to bootstrap techniques.

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 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.