Nonparametric testing via partial sorting

TL;DR

This paper introduces a novel nonparametric testing method based on partial sorting of data, specifically using the bubble sort algorithm, which is sensitive to data order and distribution, and demonstrates improved performance over classical tests.

Contribution

The paper proposes a new nonparametric test utilizing partial sorting via bubble sort, with theoretical convergence results and practical advantages over existing methods.

Findings

The empirical bubble sort curve converges uniformly to a limiting curve.

The proposed test outperforms classical nonparametric tests in various examples.

The asymptotic distribution of the test statistic generalizes the Kolmogorov distribution.

Abstract

In this paper we introduce the idea of partially sorting data to design nonparametric tests. This approach gives rise to tests that are sensitive to both the order and the underlying distribution of the data. We focus in particular on a test that uses the bubble sort algorithm to partially sort the data. We show that a function of the data, referred to as the empirical bubble sort curve, converges uniformly to a limiting curve. We define a goodness-of-fit test based on the distance between the empirical curve and its limit. The asymptotic distribution of the test statistic is a generalization of the Kolmogorov distribution. We apply the test in several examples and observe that it outperforms classical nonparametric tests for appropriately chosen sorting levels.