Random Permutations, Non-Decreasing Subsequences and Statistical Independence

Jesús E García,Verónica A González-López

doi:10.3390/sym12091415

Abstract

In this paper, we show how the longest non-decreasing subsequence, identified in the graph of the paired marginal ranks of the observations, allows the construction of a statistic for the development of an independence test in bivariate vectors. The test works in the case of discrete and continuous data. Since the present procedure does not require the continuity of the variables, it expands the proposal introduced in Independence tests for continuous random variables based on the longest increasing subsequence (2014). We show the efficiency of the procedure in detecting dependence in real cases and through simulations.

Highlights

IntroductionThe procedure is based on identifying the longest non-decreasing subsequence (LNDSS) detected in the graph of the paired marginal ranks of the observations
In this article, we use an expanded structure of the symmetric group Sn, over the set of permutations from {1, . . . , n} to {1, . . . , n}, to develop a dependence detection procedure in bivariate random vectors.The procedure is based on identifying the longest non-decreasing subsequence (LNDSS) detected in the graph of the paired marginal ranks of the observations
0.143 In Table 3, we show the performance of the JLNDn ’ s test based on the computation of the p-value (Definition 4) according to Equation (3)

Summary

Introduction

The procedure is based on identifying the longest non-decreasing subsequence (LNDSS) detected in the graph of the paired marginal ranks of the observations. It records the size of the subsequence and verifies the chances that it has to occur in the expanded space of Sn , under the assumption of independence between the variables. The procedure does not require assumptions about the type of the two random variables being tested, such as being both discrete, both continuous or a mixed structures (discrete-continuous). We have several procedures, for example, Hoeffding’s test and those based on dependence’s coefficients (Spearman’s coefficient, Pearson’s coefficient, Kendall’s coefficient, etc.)

Objectives

Methods

Conclusion